1. Being sick, he didn't go to school.

2. Because he was sick, he didn't go to school.
(Can we use both expressions? Are they the same?)

3. Being sick, he went out to play soccer.
4. Though he was sick, he went out to play soccer.
(What about these two sentences? Can we use both sentences? Are they grammatical and have the same meaning?)

2 answers

2 and 4 are correct, 1 and 3 are incorrect.
Actually, #1 is acceptable. #3 doesn't really make sense, so I find it unacceptable. "Being sick" and "Because he was sick" do mean the same thing. To rephrase #3, you might write, "Despite being sick..." or "Although he was sick..." In #4,"Although" is better than "Though".