Usage of "will" and "would"

Would is conditional/hypothetical. Will is a modal verb used to form the indicative future tense.

In both cases, a) and b) are both correct, and carry a different meaning. When would is used, it expresses a hypothetical situation, so:

What would I do without you?

... asks a question about a hypothetical possibility - that I would be without you. However:

What will I do without you?

... implies that I will in fact be without you at some time in the future, and asks a question about that factual (rather than imagined) eventuality.

In theses examples will is a more definite statement about the future while would is more hypothethical.

So in your first example, "What will I do without you?" means you are going and I am wondering what is going to happen next, while "What would I do without you?" is more speculative.

In your second example "You will always be my favorite travel buddy" is an assertion or prediction. "You would always be my favorite travel buddy" looks incomplete, and might be followed by something like "if you ever travelled with me".