Two meanings of anything.
1. He can't do anything.(=He can do nothing.)

2. Dom: He can do anything.
Tom: No, he can't do anything, but there are so many thing that he can do. (Here, he can't do anything≠He can do nothing)

Am I right?

Question

Two meanings of anything.
1. He can't do anything.(=He can do nothing.)

2. Dom: He can do anything.
Tom: No, he can't do anything, but there are so many thing that he can do. (Here, he can't do anything≠He can do nothing)

Am I right?

Answer

Sure. Seems that way.
Obviously related, but yes, very much at opposite ends of the spectrum.
positive: anything meaning any one choice of many, many, things  - versus- negative: not any one thing at all

Answer

"Anything" in all the sentences has just the one meaning. It is the -emphasis- added  that gives it the different flavor in #2:

"No, he can't do ANYTHING, but there are so many thing that he CAN do."

(In talking one would say the capitalized words with emphasis. In writing they would properly be italicized.)

Answer

Yes, that is correct