1. Yes, it's fine, and also perfectly natural, to begin a sentence with 'And..' in an informal email. 2. No, ‘I want you to come and could you bring some sandwiches’ doesn't work.This is because the structures either side of the 'and' are so different. You could make two parallel clauses, both as statements, like '‘I want you to come and I wonder if you could bring some sandwiches’. Or, if you want to keep it as a question form, you'll need to put 'Could you bring some sandwiches?', with a question mark at the end, on its own.

Many authoritative sources address the 'and' and 'but' controversy. It seems to be settled that it is permissible. I regard this as a fake controversy, right up there with the dreaded split infinitive. No, I'm not going to list all of the sources. They're easy to find.

Agreeing with the person above.... and adding. :) 1. You can do this in informal emails, but do not do it in formal writing. Technically correct writing never begins a sentence with "and" or "but." 2. While the person above is 100% right about structure, English speakers will do this anyway. You won't sound odd, at least not in the US.