Any problem whose domain is finite is Decidable or not?

Question

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 $2$:Does a single instance of "turing machine's halting problem" is decidable?
As I given above example, I think this is also Undecidable.

If I am doing something wrong then please tell me what is really mean by "Domain of a problem" and "single instance of turing machine's halting problem"? 

Comments

1) M is on finite set and on finite domain

So, it should be Recursive and also decidable. Why do u think it is undecidable?

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@reena
https://gateoverflow.in/725/gate2001-2-7
u mean single instance is a single finite string right?
But without telling this we can simply say it finite string

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yes @srestha, I mean the same. But saying this is valid or not..I am not sure about that as Arjun sir told above.

https://cs.stackexchange.com/questions/83752/any-problem-whose-domain-is-finite-is-always-decidable-is-true-or-false