Regarding query of stock

Yes.

But it will be treated as supply, even if without consideration.

With consideration: Sec. 7(1)(a)

Without consideration: Sec. 7(1)(c) read with Schedule-I
Supply between related persons made in the course or furtherance of business. Brothers will be related persons under GST, since they are members of the same family.

Yes, the person can transfer his shares to the other person.