Yes, you are right. To solve for a variable means to put it on one side by itself, in terms of the other variables or values. To move something to the opposite side you do the opposite operation. Then you do the same thing to both sides of the equation.
To solve a formula or equation for a specified variable means to rearrange it so that the variable appears alone on one side of the equal sign. I got this right on usatestprep.
Yes, the blank numbers that can be written as a product of blank factors are called the prime factorization of a number. It can also be composite numbers where it can be written also as the product of prime factors.