Is it decidable when a TM M gets another as inputs and checks if it fullfiills certain property?
That definitely isn't enough. The set of Turing machines that always halt is countable (being a subset of the countable set of all Turing machines), but very not decidable.
Don't overcomplicate: If you are given a Turing machine, any property of the machine as such (number of states, number of transitions, is it deterministic, ...) that can be checked by a deterministic algorithm is obviously decidable.