Is halting problem computable for particular inputs/assumptions
is halting problem computable by program P for all other programs used as input but P itself?


No. Consider the infinite sequence of programs $P_1, P_2, \dots$, where $P_i$ is "Move the head $i$&nbsp;squares to the right, then $i$&nbsp;squares to the left, then do exactly what $P$ would do." Every one of these programs produces exactly the same output as&nbsp;$P$ for every input (including looping forever if $P$&nbsp;loops forever), but they're all different programs.

And you can't get around this by only requiring $P$ to work on programs that aren't functionally equivalent to itself, since that property is also undecidable. Perhaps the problem would be decidable when restricted to those instances (though I suspect it wouldn't) but the set of instances is undecidable, so you couldn't actually perform the restriction.