Question

1. I got out of my car to check the engine.

2. I got off my car to check the engine.
(Are both the same? In #2, what is the part of speech of 'off'? Is 'off' an adverb or a preposition?)

3. I got out of my car to check the engine.

4. I got out of my car to check out the engine.

5. I got out of my car to check on the engine.
(Are the three the same? What is the function of 'out' or 'on'? Are both adverbs?)

Answer 1

1 is correct, but 2 is not.
"Off" means you were on top of it and then you got off!

3, 4, and 5 are okay; 3 and 4 mean the same thing, but 5 is slightly different.

To check out, to check on, to get out, to get off -- all those are phrasal verbs:
http://grammar.ccc.commnet.edu/grammar/verbs.htm#phrasal