There are different versions of the Halting Problem:
1) $\{ P \mid\text{there exists $i$ such that $P$ halts on $i$} \}$,
2) $\{ (P,k) \mid \text{there are less than $k$ inputs that $P$ halts on} \}$,
3) $\{ (P,k) \mid \text{there are $k$ (or more) inputs that $P$ halts on} \}$, and
4) $\{ P \mid P \text{ halts on } 0 \}$.
(1) is the standard problem in most books I've read. The proof that (1) is recognizable is just: Run P on i and accept if P halts.
Can someone show me the proofs for the other 3 (assuming they are recognizable...)
The proof for 3) is simple: enumerate all inputs $(w_i)_{i\in\mathbb N}$ and run $P$ on each of them, using the standard trick (run $1$ step of $w_1$, then $2$ steps on $w_1$ and $1$ step on $w_2$,...). As soon as $P$ halts on $k$ inputs, accept $(P,k)$. As a consequence, 2) is not recognizable: it if were, one could decide the language $\{(P,k)\mid P \text{ halts on exactly } k \text{ inputs}\}$, and then we could decide the language $\{(P,w) \mid P\text{ halts on input } w\}$, which is a standard version of the halting problem.
Recognizability of 4) is trivial, run $P$ on $0$ and see what happens.
The proof for 1) is not quite was you say: you don't have $i$, so that you can't run $P$ on $i$. But 1) is just the specialization of 3) with $k=1$.