1. I got up early getting ready for the game.

2. I got up early while I was getting ready for the game.

(Is #1 grammatical? Does #2 mean #1?)

3. I helped my friends with teaching them math.

(What about this one? Is this also grammatical?)

4. Eating vegetables is good for you.
5. Eating vegetables and eating fruit are good for you.

(Do we have to use 'is' or 'are'?)

6. I'm sorry that I am late.
7. I'm sorry for being late.

8. I'm sorry that he is late.
9. I'm sorry for his being late.

10. I'm sorry that he was late.
11. I'm sorry for his having been late.

(Is each pair the same and grammatical?)

3 answers

Rather than have some form of the word "get" in a sentence twice, here's a rephrasing for 1 and 2:
I woke up early to get ready for the game.

3 -- I helped my friends by teaching them math.

4 and 5 -- You have #4 right. #5 should also have "is" instead of "are." The subject is "eating" which is singular.

6 - 11 are all correct, and yes, each pair means the same thing. The phrasing in #11 is rather awkward, though, and rarely used.
Thank you.

3. I helped my friends with teaching them math.

(Is this one ungrammatical?)
It's not ungrammatical; it's just not the correct preposition to use in that phrasing.

You could also say, "I helped my friends with their math." That would mean the same thing, but with slightly different words.