Can the Turing machine be classified e.g. as a Mealy machine? Why not? Can a Turing machine be input to another Turing machine without complication like halting problems?
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For your first question, consider, for example, Wikipedia's formal definition of a Mealy machine. Your question is precisely equivalent to "can a Turing machine be described in terms of that definition?" Looking at a formal definition of a Turing machine would be instructive.
For your second question, you may want to research Universal Turing Machines. They are a formulation of nearly exactly the idea of "inputting a Turing machine into another Turing machine", and any reasonable resource that discusses those more than in passing will certainly help give you some idea of how the halting problem relates to them.
This sounds awfully like some homework, so I'll refrain from giving a more detailed answer, but considering inputting a Turing machine: