Is this correct?
The person with whom I'm doing the project should be here soon.
If it is, is with always a dative preposition (like mit in German)?
When "who" is the object of the preposition, as in this case, it becomes "whom"; granted, this is by now vestigial and often ignored in informal conversation. You'll often hear people say things like, "Who should I give this to?" It would be correct to say "Whom should I give this to?" and misguided fussbudgets will insist you render it as "To whom should I give this?" But almost no one bothers with that these days. Note that reversing the word order makes the incorrect grammar stand out: "I should give this to who?" That's because there is now a direct apposition with the preposition and its object. Most careful speakers will use "to whom" in that context.
You can remember when to use "who/whom" by substituting "he/him" in the sentence. You wouldn't say "I'm doing the project with he," you would say "I'm doing the project with him." So it's obvious that whom is the pronoun you would use here, not who.
A further word about German/English prepositions. In German some prepositions can be dative or accusative, depending on whether they indicate motion or placement towards or up to a location. This not the case (no pun intended) in English. In English, the object of the preposition always takes the "prepositional" case. Note that there are not nearly as many inflectional changes or pronoun substitutions in English as in German. The point is, German is not necessarily useful for analogizing English constructions.
Here is a really easy way to deal with case and prepositions:
If the the preposition is directly modifying the noun, then the noun is always* accusative/dative. And, since accusative and dative forms both look like whom, you know it should always be whom (if you are using whom at all).
*There is one major exception to the prep. phrase rule: of sometimes takes the genitive case ("friend of his"); however, this never shows up with who (there is no "friend of whose", just "whose friend").
It is worth also noting that whom is falling out of use, and so who could theoretically be used everywhere when speaking in a register that doesn't use whom.
In your case, the person is the object of the sentence, while I'm is the subject. Even though the sentence has a questionable structure, it's correct.
This article by Mignon Fogarty ("Grammar Girl") does a remarkable job explaining when to use who and whom.
When you’re trying to figure out whether to use who or whom, it helps to know the difference between subjects and objects because you use who when you’re referring to the subject of a clause and whom when you’re referring to the object of a clause.
"The person with whom I'm doing the project should be here soon" is grammatically correct. However, it's not the only way, or necessarily the best way of expressing this idea.
There is no such thing as a “dative preposition” in English.
English has a very limited case system. Case is only marked on some pronouns, not on any regular nouns. And only two forms are distinguished: one for subjects (which could be called subjective or nominative case) and one for various other functions (which could be called objective or accusative case). I’m not counting forms like “my”, which are sometimes called genitive case, as part of the case system, since they are generally only used as determiners.
The object (also called the complement) of a preposition is always in the objective case.
The only complication I can think of is that some words, such as than, may be treated by some native speakers as prepositions (in which case a following pronoun must be in the objective case), and by others as conjunctions (in which case a following pronoun may be in the nominative case), e.g. “I run faster than he” vs. “I run faster than him.”
Also, I think Kosmonaut was right to point out the "(an) X of mine" construction as another possibly confusing case. I don't know if I would say that "of" governs the genitive case in this construction; when I tried to find an analysis of its grammatical structure, I came across various explanations. But it definitely looks at first sight like a case of a preposition governing the genitive case in English. You can find a few examples online of "of whose" being used in a context like this; e.g. "Baltazar told us of a man he knew, a chicken of whose had been stolen by a puma" (’Tambo: Life in an Andean Village by Julia Meyerson).
Native English speakers naturally use distinct forms for the subjective and objective case of the personal pronouns I/me, he/him, she/her, and they/them. Native speakers don’t naturally use distinct forms for the interrogative or relative pronoun who. Rather, the form who is used in both subjective and objective contexts; it’s treated as an invariable pronoun like you or it.
However, in formal writing or speech, the form whom may be used in objective-case contexts where who would be used in colloquial speech.
The only circumstance where whom is required (rather than an optional replacement for who) is in a certain construction that doesn't really occur in colloquial speech.
Your sentence actually uses this construction. It is called "pied-piping" of prepositional phrases: it occurs when a prepositional phrase containing a relative or interrogative pronoun is fronted as a whole.
In colloquial English, the preposition is usually stranded (left in place) rather than fronted in clauses of this type. This also obviates the need for any relative pronoun at all:
The person I'm doing the project with should be here soon.
By the way, by “colloquial” I just mean everyday spoken English. The preceding sentence is not markedly colloquial; it doesn’t sound informal and it would be appropriate to use it in spontaneous speech even in a quite formal setting. In fact, it's hard for me to think of situations where it would be inappropriate. Maybe in formal legal documents.
The person with whom I’m doing the project should be here soon.
As thesunneversets mentions in a comment, this sentence with pied-piping sounds pedantic and fairly “stuffy” to many people, although it is considered to be grammatically correct. It’s hard for me to think of situations where it would be more appropriate than sentence 1; perhaps if you’re talking to someone you know is a snob about this kind of thing, or writing a very formal document.
It sounds somewhat unnatural, and markedly higher-register to use a fronted prepositional phrase containing a relative pronoun like this. John Lawler has mentioned in a comment that "whom is required only when it is the object of a pied-piped preposition."
The person who/that I’m doing the project with should be here soon.
If you strand the preposition, it sounds best to me to omit the relative pronoun, but it is also possible to use an explicit relative pronoun in this context. If you do, either who or that sounds OK to me. (Some people think "that" cannot be used for animates, but this is a misguided viewpoint. Still, it may be true that "who" would sound better than "that" in this sentence.)
?The person whom I’m doing the project with should be here soon.
To me, it doesn't sound good to use whom at the start of a relative clause with a stranded preposition. If I wanted to use whom, I would prefer to use pied-piping as well, as in sentence 2. John Lawler seems to feel the same way about this as I do; however, Janus Bahs Jacquet pointed out in a comment that sentences like 4 sound natural to him, so it seems not everyone agrees about this. Even the most pedantic grammar "authorities" agree that preposition-stranding is grammatical, so this sentence can't be said to have objectionable grammar (unless you take the unconventional view that the word "whom" is objectionable). See the answers to the following question: Prepositions at the end of sentence and whom.
*?The person with who I’m doing the project should be here soon.
I can't imagine any circumstance where this sentence would be preferable to 1, 2, 3 or 4. I actually would call it ungrammatical, although some people might disagree with me about that (see Janus Bahs Jacquet's comment on the following post: "...four others, one of whom responded." Is “whom” correct here? Can I use "who" instead?, and the following examples found on Google Books: a, b).
Basically, what goes for relative pronouns also goes for interrogative pronouns.
"Who am I doing the project with?" is good normal colloquial English. "With whom am I doing the project" sounds more formal, at least a bit unnatural, but it's OK also. "Whom am I doing the project with" sounds funny to some people (including me) but OK to others; "With who am I doing the project" sound pretty bad and is never advisable.
With questions, there's another structure to be aware of. While questions in English normally exhibit fronting of the question word, there are also "Wh-in-situ" questions where the question word occurs in the same place as the corresponding noun phrase in a corresponding declarative sentence. This is common in "echo questions" where a speaker indicates disbelief by repeating a declarative sentence with a word replaced with an interrogative pronoun, as in the following short dialogue:
In questions of this type, the pronoun who would be used in normal/colloquial speech as the object of with, and whom can be used as a more formal variant.
Examples from Google Books of "with who" in contexts like this:
You Want Me to Work with Who?: Eleven Keys to a Stress-free, Satisfying, and Successful Work Life ... No Matter Who You Work With, by Julie Jansen (Note also that the subtitle is an example of the "fronted who as object of stranded preposition" construction)
"Tell all the Scandal, whether false or true, / Absurdly vain of knowing who's with who" ("The Veil Removed", in A collection of English songs, with an appendix of original pieces, edited by A. Dalrymple, the original pieces by James Dalrymple, 1796)
With, like any other English preposition, takes an object/complement in the objective case.
This doesn't mean it can't be followed by who. In modern English, the pronoun who can be used as either a nominative case form or an objective case form.
The pronoun whom is only used as an objective form; it's optional in most contexts. However, it is strongly preferred over who in one particular circumstance: when it occurs in a fronted prepositional phrase such as "With whom are you going?" or "The man with whom I was speaking". It would sound bad to use "with who" in these contexts. However, in colloquial English this construction with whom is avoided by stranding the preposition: "Who are you going with?" or "The man I was speaking with".
"Whom" is for object ; "who" is for active subject.
It should work with "Me/him/us/them" : object - whom ; or "I/he/us/they" linked to who. As all we know, it's not so simple....
Why ? because sometimes in forms like : "Do you want to go there with me ?" or "He & Josh will buy it" we hear instead "with I" & "Him"... Yes it's wrong but you know I'm right, we say it ! So you'd have to correct before, if you'd like to know whether "whom" or "who" would match with "Him" or "He" etc.
However, we also have an other problem with : "It's me !" to the question "who's that ?" Here "who" is correct but our common ansmwer"me" is just not ! Yet as we say it, it's good (in the UK they say like us : "me")... Indeed, "It's I !" would sound precious but it's the correct answer.
Take also this : "Who(m) did you say was an idiot ?" Here you may say either whom or who. Who is active for idiot but also you can use whom because it's him you call an idiot.
Other way : "Where is she now ?" "I looked in the trunk ; it's her !" "Who ?" "Her : Samantha !" We see above that "her" we all use should be in fact "she" but we don't... That explains why it is not so easy to see how always go who & whom without prejudice of the fact that we use the sole "who" for both "whom" & "who".