HINTS:
(1) Let $\Bbb X=\{0,1\}$ with the discrete topology, and let $f:\Bbb X\to\Bbb X:x\mapsto 1-x$. (Yes, this is a metric space; $d(0,0)=d(1,1)=0$ and $d(0,1)=d(1,0)=1$ is a metric.)
(2) It’s true; try to prove it. You may find this useful.
(3) A closed subset of a compact space is compact. What do you know about the continuous image of a compact set?
There aren’t really any tricks. (2) and (3) are standard results or immediate consequences of standard results, so with a bit of experience one simply knows them. Someone with a good intuitive feel for compactness would probably guess that they’re true, but I’d guess that almost everyone learns the standard results before developing that good a feel for the property.
The example that I suggested for (1) came from a basic strategy for approaching any result when you don’t know whether it’s true or not: look at some simple examples.