What allows the bare infinitival in all of your example sentences isn't "does", but the specifying "be". It is used to specify or identify the referent of "what" or "the thing". That "be" appears in these sentences as "is". In all your examples, "to" is optional.
What they do is [to] make trouble.
What a good worker does is [to] get to work ...
The first thing I do in the morning is check my mobile phone
This is perfectly grammatical and perhaps the most common. When the subject has a form of "do", the infinitive is used. You can put "to" there, but it is optional.
The first thing I do in the morning is checking my mobile phone
The sentence above is not grammatical. It'll only be grammatical ...