"Any problem whose domain is finite is always Decidable", is true or false? Question no:$1$


  "Any problem whose domain is finite is always Decidable"  


lets take a TM,$M$ and finite domain of problem i.e. finite set of strings for eg.  {$a,abaa,bba$}, Now the problem "whether $M$ accepts these set of strings or not is decidable or not?"  

What I think is, this is Undecidable problem because, if $M$ loops forever any of these string then we not determine whether turing machine accept it or not.so according to this logic above statement is false.

Question no:$2$

Does a single instance of "turing machine's halting problem" is decidable?
As I given above example, I think this is also Undecidable.
It's not so clear what is "a problem whose domain is finite". Let me assume that what is meant by this is a finite language. Every finite language is decidable. The Turing machine compares the input to each word in the language, accepts if it finds a match, and rejects otherwise.

This answers your first question. Your second question is a special case of the first one.