Yes, they mean the same thing and are correct. -Side note- If you were in the states and someone asked you, Do you play soccer? You would respond with “Yes, I play soccer.” or “No, I don't play soccer.” You would not say “I am able to play soccer” or “I can play soccer.” It would infer that you are handicapped in some way or need permission.

Yes, they are basically the same. It is more common to say "I can play soccer."

Yes, you can say I’m able to cook. Nothing wrong with that.

What about ‘I’m able to cook’ then?

"I can......" usually means you are competent in that subject. eg. "I can cook" or "I can draw" Anybody can cook or draw but you would only say the above phrases if you had some skill in those areas.