1. If " how's " in the sentence "How's it feel?" means " how does " so that there is no grammar mistake.
There is no grammar mistake. When we say 'How's it feel?', this is a contraction of 'How does it feel?'. The grammar is correct, but the contraction is a little non-standard. As you know, the standard contraction " 's " in written English is either 'has' or 'is'. However, in informal spoken language we sometimes contract 'does' to 's' as well. For example, 'What does he do?' may sound like 'What's he do?'. Again, this is not a grammar error. When we say this, the part of our minds which constructs the grammar is saying 'does', not 'is'.
2. How's it going ? I know it is " how is " ,so" how's "might mean both "how is" and" how does " ,right ?
No. Absolutely not. 'How's it going?' is the contraction of 'How is it going?'. This is a present continuous form. Nobody says 'How does it going?' unless they are a foreign learner making a bad grammar mistake.
3. 'How's it feel ?' and 'How does it feel ?' ,which one is most commonly used in spoken English ? (I know usually we say "how's it feel to do something ...)
In spoken English, we would not usually say a clear stressed 'does'. We would weaken the auxiliary to /dəz / or reduce it to /dz/ or just /z/.