1 should be this:
He is trying to cut wood with an axe.
In 2, both of the first expressions are correct. The wording "a wood" and "two woods" are not correct in normal usage (except in highly specialized references to golf clubs).
1. He is try to hit a piece of wood with an axe to cut it.
(Is this correct?)
2. a piece of wood, two pieces of wood
(Do we use these expressions? What about a wood, or two woods?)
2 answers
1. Could also be: He tries / He is trying / He does try (Present Tense) rather than the Present Progressive.
2. If you have mahogony wood and birch wood, you could then say: the two woods
Sra
2. If you have mahogony wood and birch wood, you could then say: the two woods
Sra