Difference between homotopic and homeomorphic
Homeomorphic is stronger than homotopic. For instance, a deformation retract $A$ of $X$ is homotopic, but not necessarily homeomorphic. (Homeomorphic is literally isomorphic, you have a bijective "bicontinuous" map) Of course, deformation retract is a stronger condition than homotopic too.

Fundamental group of the real projective plane
Is NOT SIMPLE!! In fact: $$\pi(RP^2) \approx \langle a | a^2 \rangle$$ Link to understand why it's not simple that didn't really help; but the idea (I think) is that $S^2$ is a covering space of $RP^2$, so you can lift paths in $RP^2$ into $S^2$. The base point in $RP^2$ corresponds to two antipodal points in $S^2$. Non-trivial paths $\gamma$ get lifted to a path in $S^2$ that starts at one point and ends at the antipode. For some reason $2\gamma$ is trivial when it gets lifted though...

Finding continuous maps
It's sometimes easier to take an indirect route. For instance; you'd think that there's no continuous surjective map from $S^2$ to $S^1$, but you'd be wrong. Just project $S^2$ to a line, (i.e. just take the $z$ component), and now wrap the line around the circle. Composition of continuous maps are continuous. Be indirect. Be holy.