What are the more precise relations between (a) projective space, (b) quotient space and (c) the base manifold under certain fibration?
- (1) Can every projective space (e.g. $\mathbb{RP}^n$, $\mathbb{CP}^n$, $\mathbb{HP}^n$, and many others) can be constructed as a quotient space?
(In some cases, there are the quotient space of spheres over spheres. Say $\mathbb{RP}^n=S^{n}/\mathbb{Z}_2$, $\mathbb{CP}^n=S^{2n+1}/S^1$, $\mathbb{HP}^n=S^{4n+3}/Sp(1)=S^{4n+3}/S^3$, yes?)
(2) Can every quotient space can be constructed as a projective space?
(3) It looks that (b) and (c) all involve a relation of modding out or quotient out certain space (e.g. the fiber for the later case (c)). How are (b) and (c) related, or are they exactly the same? Or which one is more general?
There aren't that many other projective spaces other than the ones you've already listed. Some people might also include the "octonionic projective plane" (which can be constructed as a quotient space) and its generalizations, but this might not be conventional depending on the setting. Of course, you can also easily define projective spaces over any field (and if you know some algebraic geometry, over any commutative ring) using the same quotient definition.
Unless I'm misunderstanding you, this question doesn't really make sense. Every nonempty space is a quotient space (e.g., take $X / \{x\}$ for $x \in X$).
In a fiber bundle $F \to E \xrightarrow{p} B$, say with $p$ surjective, $B$ can be identified with the quotient $E/\sim$, where $\sim$ is the equivalence relation $x \sim y$ if $p(x) = p(y)$. In fact, this is true for any onto map $p$, but in the case of a fiber bundle, each equivalence class is equipped with a topology that makes it homeomorphic to $F$.
The point I was trying to make in the comments is most easily seen in the case of topological groups. Let $H$ be a subgroup of $G$, and consider the expression $G / H$. If one thinks purely topologically, then by definition this is the space $G$ with just the identity coset $H \subseteq G$ collapsed to a point. However, what most people mean by $G / H$ is to collapse every coset of $H$ in $G$ to a point, so that $G / H$ is the space of cosets of $H$. The "quotient" in a fiber bundle is more like the latter and not the former.