Best way: Could I give you some advice? OR (strictly correct) May I give you some advice? OR Would you mind if I gave you some advice? According to the strictest rules of etiquette, you should use "may I", rather than "could I" when asking permission to do something. However, in actual practice, this rule is observed only some of the time in spoken English.

Grammatically it is correct. Can is one of the modal verbs (can, could, would, should, shall, will, might, must and may). The way you figure it out is through subject-auxiliary inversion. Just do it backwards and forwards as a question and a statement: "Can I give you some advice? or "I can give you some advice." The problem here is that they almost always sound correct as a verb in a sentence or question, because they express opinion, or ability to do things. In this case because you are asking permission, you should use "may I" as in "May I offer you some advice." However, you can say "can I give you some advice." It would not sound weird to a native speaker. "May I" would be a more polite to ask. Does this help? Or should I do this again? I can, if you would like. You hear them a lot in English; look how many I have used! Just listen for them, and you'll figure out where people often use them when speaking. In this case, you should use "May I."

It's perfect fine. "A piece of advice" works very well, too. It really depends who you are talking to. Some might welcome the advice. Others might be offended.

You can say either. Can I give you some/a piece of advice? Having said that, it's more common to hear 'let me give you some advice', or even 'a word to the wise:...'/ 'just a piece of advice:...'. Other options include 'I'd suggest/advise...', 'I advise you to...' 'You might...' Any of these or variations on the same would all be perfectly acceptable. Amongst friends, I personally would say 'I advise you to...' and with more formal company, I might say, 'I'd suggest...'. :-)