On the computable function of a problem that halts
No, it is not.

In general, to help you think about this kind of problem, I recommend you work through some examples.  Keep them as simple as possible.

Here is an example that illustrates why the answer is no.  Suppose $P$ is the following program: "if the input is 1, then halt immediately; otherwise enter an infinite loop".  Suppose $F(1)=2$.  Then $P$ halts on the input $i=1$, but it does not halt on the input $F(i)=F(1)=2$.  So, the behavior of the program $P$ on one input tells you nothing about its behavior on a different input.