We always start with "given" when it comes to proofs in geometry, and proofs in general. Think of it like starting to build a house. Someone goes out and buys the material, which is the "given" that you start to work with.
When we cross two lines to form an X shape, the angles opposite each other are known as vertical angles. They don't need to align vertically to be considered as such. Angles 2 and 3 fit the description. Vertical angles are always the same measure. I'm not sure why your teacher wanted to emphasize "1st occurrence" and "2nd occurrence". That part is irrelevant in my opinion.
See item #2 above.
Alternate interior angles are angles inside the parallel lines, and they are on alternating sides of the transversal cut. One pair of alternate interior angles is angle 3 and angle 6.
The transitive property is the idea of chaining equations together. If we know that A = B and B = C, then we link them together to conclude that A = C. You can think of it as if you started in town A, moved to town B, then ultimately ended up in town C. So therefore you've gone from A to C.
