Prove if a property of a Turing Machine is decidable or not, how can I do it?
To show that a language is decidable, you need to provide a description of a total Turing machine, i.e, on all input, the Turing machine should halt and accept or halt and reject. 
Take the example of the first problem, we can build a total Turing machine, that on input $M$(where $M$ is a Turing machine) runs it all inputs of length at most 100. Note that if $M$ halts on all inputs within 100 steps, then it must halt on all inputs of length at most 100 within those many number of steps. If $M$ does so, our constructed Turing machine halts and accepts, else halts and rejects. Since  we have to check only on a finite number of input strings, the constructed machine always halts and hence is total. 
As for the second problem, the property of accepting all strings with an 'a' in them is a non-trivial one for recursively enumerable sets($L(M)$ or the language recognized by a Turing machine $M$ is an R.E set) and hence from Rice's theorem, the language of deciding recursively enumerable sets with this very property is undecidable.