For 1 and 3, yes. (For 3 assuming both are positive, technically one positive would suffice but then it is essentially reduced to 1, while of course both negative will not work.) 

And, note that this is only interisting for natural coefficients, with integers you can write any integer whatsoever, and this is very classical, not even sure how to call this, a direct consequence of Bezout's identity maybe, or an immediate consequence of results on linear diophantine equations.

For 2. The main keyword (with natural coefficients) here is Frobenius problem, but mainly for more than two numbers. Also Coin Problem, Postage Stamp Problem or Chicken McNuggets Problem are common more playful names. See http://en.wikipedia.org/wiki/Coin_problem

The solution for 3 is due to Sylvester.

If you have more than two coprime natural numbers $a_1,\dots , a_k$ for integer coefficients you still can write everything. But for natural number coefficients you can only write everythong starting from a certain number $F(a_1,\dots , a_k)$ on. To determine the optimal value here is *hard*. This is the Frobenius Problem.

It was solved in a certain sense recently for three numbers (by Fel, I think, I will check later).