1. Because Tom got ready for the game, he got up early.
2. Getting ready for the game, he got up early.
(Does #1 grammatical? What about #2? Are both the same if they are grammatical?)
3. He ran away seeing me.
3-1. He ran away when he saw me.
4. You will find the house if you turn
to the right.
4-1. You will find the house turning to
the right.
5. I still don't believe it though I
admit what you say.
5-1. I still don't believe it admitting
what you say.
6. I can't help you as I don't have
enough money.
6-1. I can't help you not having enough
money.
(Is each pair grammatical and the same?)
1 does not make sense. It would be better to write, "Tom got up early to get ready for the game." BUT there are two instances of "got" in that sentence. That's what is making it not work well. So try this: Tom woke up early to get ready for the game.
2 is OK, but still has to instances of the word "get" in it.
3 is not good, but 3-1 is fine.
4 is fine, but 4-1 is not.
Both of the 5's are awkward and not quite right. Better would be this: I still don't believe it even though I admit what you say is correct.
6 is fine; 6-1 is not.