- My hometown is a great place to live.
- My town is a great place to live in
This is an excellent question. The answer, interestingly, has nothing to do with Dublin. It has absolutely nothing to do with the verb live. It also has nothing to do particularly with the prepositions in or at either. It has everything to do with the noun place.
The noun place is not a normal noun. If you replace place with most other nouns in English, you'll find that it isn't in fact grammatical to drop the preposition at the end of the infinitive in sentences like these:
- *Dublin is a great city to live. (ungrammatical)
- *This is a great restaurant to eat. (ungrammatical)
- *This is a great hotel to stay. (ungrammatical)
- *This is a horrendous zone to have to live. (ungrammatical)
Compare those to:
- Dublin is a great place to live.
- This is a great place to eat.
- This is a great place to stay.
- This is a horrendous place to have to live.
Notice as well that if we want to add a Locative Complement or Locative Adjunct (read Adverbial) to a normal sentence, we normally need a preposition phrase:
- Let's eat in some restaurant.
- He was staying at some hotel or other.
We cannot drop the prepositions from these sentences:
- *Let's eat some restaurant. (ungrammatical)
- *He was staying some hotel or other. (ungrammatical)
But, especially when it occurs with the determiner some, noun phrases using the word place can most often occur on their own without any preceding preposition:
- Let's eat some place.
- He was staying some place or other.
With relative clauses, we also see the noun place behaving in an exceptional way. Relative clauses have an antecedent, which is the phrase that is being modified by the relative clause, the thing that we are giving more information about. Some nouns, in particular, the word place can have relative clauses where the gap in the relative clause represents a locative phrase of the type that would normally be represented by a preposition phrase within the relative clause. This is not normally possible unless the relative clause uses the relative word where. However, when place - or a small number of similar words - is the antecedent noun, it is possible. Consider the following sentences which are grammatical when the antecedent is the word place, but not if it is another noun:
- That's the place we ate last time
- Dublin is the place I want to live.
- That's the place we saw that incredible film.
- This is the place they kept the prisoner.
Compare those with the following:
- *That's the restaurant we ate last time. (ungrammatical)
- *Dublin is the city I want to live. (ungrammatical)
- *That's the theatre we saw that incredible film. (ungrammatical)
- *This is the dungeon they kept the prisoner. (ungrammatical)
And then consider these, which use the relative preposition where:
- That's the restaurant where we ate last time.
- Dublin is the city where I want to live.
- That's the theatre where we saw that incredible film.
- This is the dungeon where they kept the prisoner.
The Original Poster's questions
Normally when we drop the word in after the word live it is because of some occurrence of the word place. This word may occur as an antecedent,as a word that is being modified by some clause that live occurs in, or it may be part of a noun phrase functioning as a Locative Complement of the word live. In most circumstances, it is not because of any special relationship with the word live itself. Indeed, this will occur with many other verbs. Very often the results when we drop the preposition, are better than the results when we don't:
- London is the best place I've lived.
- London is the best place I've lived in.
Personally, the first example above is more natural than the second. However, using sentences like this will often convey an informal style. When a more formal style is required, we may be better off substituting the word place for a different noun and using a regular preposition phrase as a complement.
Regarding the second question
It's the East End" that I lived in all of my life.
It's the East End in which I lived all of my life.
It's the East End which I lived in all of my life.
If the relative words which or that are used in such relative clauses then they will be interpreted as representing noun phrases in the relative clauses themselves. They won't be interpreted as standing in for preposition phrases or locative phrases in general. For this reason we still require the preposition in to meet this requirement. As the Original Poster has shown, where does indeed represent a locative phrase within the clause itself and, for this reason, no preposition is required.
So I've given a vague description of what can happen with the word place. Exactly why does it happen this way? I don't know anyone who knows. One theory that we could put forward is that place is becoming more and more preposition-like. It is taking on more and more prepositiony qualities. Perhaps one day it will actually split in two and may become a locative preposition in its own right. But - that's just an interesting theory. Nobody knows!