When adding mixed numbers, do the denominators have to be the same?

Yes the denominators have to be the same in order to add. If the denominators are different, then you need to find a common denominator and then solve the equation.

If you have ten monkeys sitting in the tree and then two more jump up into the tree, how many monkeys are now sitting in the tree?

You are going to add the two sets of monkeys together, which would be 10+2. There are now 12 monkeys sitting in the tree

When solving an equation, what is the order of operations for equations?

When determining the order of operations you first solve within grouping symbols, then you solve exponents, next you solve Multiplication/Division, and lastly you solve addition/subtraction in the order of appearance.