Some basic questions on halt and move in Turing machines
To answer any kind of question like this, you need to choose one of the standard definitions of Turing machines (there are several but they're all essentially the  same) and prove that adding the feature you want doesn't increase the computational power. You do that by showing how to simulate the feature using the standard machine.


  
  In general does a TM only have only one halting-operation? (i.e. can only one operation promote the halting or can more than one operation do that?).
  


It doesn't matter. If you want three halting states and I insist that there can be only one, then have three states called $h_1$, $h_2$ and&nbsp;$h_3$ and design the transition function so that, if the machine ever enters one of those, its next transition is to $\mathrm{HALT}$.


  
  Can the tape-head also stop at a fixed position? (i.e. not move its head). Instead of the rules 0 (move left) and 1 (move right), it can also 2 (not move anywhere) or even jump two units to the left or right?
  


Again, it doesn't matter. If I insist you must move left or right, you can move one step left and then move back to the right; you can move two steps to the right by moving one step, twice.


  
  Can the tape-head move more than one unit to the left or right?
  


Even that doesn't affect things: random-access Turing machines have an "address tape" onto which you can write a number and a special state that causes the head to move straight to the tape cell indexed by that number. Again, same power.

Multiple tapes, inserting and deleting characters, two-dimensional (or more!) tapes. Almost anything you can imagine makes no difference, and proving these things are standard exercises in computation theory textbooks.