Can I use contractions when I want to apply them to multiple nouns?
I want to say this:
Sally and I will go to the beach.
Can I say this?
Sally and I'll go to the beach.
I have noticed in my own speech that, at least in some cases, when the contracted form has a pronunciation which depends on both of the uncontracted forms, a coordinated structure, or relative clause construction, blocks contraction. It is a peculiarly complicated constraint, so my statement of the constraint may be unclear, -- please consider it together with the following examples.
Some pronouns have special single syllable contracted forms before an auxiliary verb or the 's possessive which are not indicated in the spelling. The special contracted form is missing the final off glide which would ordinarily be present at the end of the pronoun. I doubt that all American English speakers accept these special contractions. I'll use an extra apostrophe to show where this off glide would have been, if it had not been lost. For instance, all of the following are possible:
He will be there. He'll be there. [sounds like "heel be there"] He''ll be there. [sounds like "hill be there"] He has been there. He's been there. [sounds like "he z been there"] He''s been there. [sounds like "his been there"]
I believe that these special contracted forms do not occur with non-pronouns or pronouns which come at the end of constructions within the subject (like coordinations):
*Mary''s been there. She or he's been there. *She or he''s been there. [would sound like "she or his been there"] *Some man taller than he''s been there.
I believe that in most English dialects the contraction "'ll" for "will" is not productive, unlike "'s" (both for "is" and for the possessive). Have you heard people saying "The president'll do what is right." or "Trump'll get kicked out of office."? Google has 1 million results for "Trump will get" while only 3 thousand results for "Trump'll get". The BNC corpus has 334 occurrences of "the government will" but 0 occurrences of "the government'll" (you must put a space before the apostrophe to search in BNC).
It is more difficult to analyze your kind of phrases by a simple search. It is true that "you and I'll" is in use, and here are the matches in BNC:
You and I'll have dinner together then. But how's about a drive round the island -- just so that you can see for yourself what it's like.
You and I'll have to start passive exercises.
'I'll have a glass of rum,' said Black Dog, 'then you and I'll sit and talk like old friends.'
But I suspect that this is just an instance of the closed category of such constructions, and there is generally no productive use of the contraction except for "is". This post supports this claim for "are".
Here is a bit of evidence from the BNC. Searching for "NAME and I 'll" yields 42 occurrences, but all of them (except possibly the last) clearly have "and I'll" beginning a new statement. The last occurrence is from a transcript that seems rather poorly recorded, so basically there is not one occurrence of the kind of construction you asked about.
Furthermore, searching for "NOUN and NOUN 'll" yields 6 occurrences of "mum and dad'll" where "mum and dad" is the subject. Note that "mum and dad" is probably an idiomatic phrase, because it occurs 486 times in the BNC while "dad and mum" occurs only 11 times. Besides "mum and dad'll", there is 1 occurrence of "mummy and daddy'll" and 1 occurrence of "mothers and fathers'll", and apart from these there were no other matches!