**Edit:** I have added some definitions and details to my answer.

In the most general form I can find, your third question is a consequence of two results regarding cell-like maps and fine homotopy equivalences. It is closely related to and makes use of Chapman's celebrated theorem on topological invariance of Whitehead torsion.

Firstly, a *cell-like map* is a map which is proper, and whose fibres are all cell-like spaces. A compact metric space is called *cell-like* if it is "shape contractible" in the sense of having trivial (Borsuk) shape. Secondly, a map $f:X\to Y$ is a *fine homotopy equivalence* if, for any open cover $\mathcal{U}$ of $Y$, $f$ has a homotopy inverse $g$, and corresponding homotopies to the two identity maps which both follow paths never leaving some open in $\mathcal{U}$. Finally, recall that ANR abbreviates *absolute neighbourhood retract*. Importantly, and to avoid piling up qualifiers, all ANRs in this answer will be implicitly assumed metric and separable. A good example of a (metric separable) ANR is a locally finite, countable CW-complex.

The main theorem in the article "Mappings between ANRs that are fine homotopy equivalences" by William Haver implies that any cell-like map between locally compact ANRs is a (proper) fine homotopy equivalence. In that article, a cell-like map would be called a proper $UV^\infty$-map instead.

On the other hand, corollary 3.2 in the article "The homeomorphism group of a compact Hilbert cube manifold is an ANR" by Steve Ferry implies that any proper fine homotopy equivalence between locally compact ANRs is a simple homotopy equivalence. The proof of this result by Ferry uses the theory of Hilbert cube manifolds, via the following facts: (1) a proper fine homotopy equivalence between Hilbert cube manifolds is approximable by homeomorphisms (as proved by Ferry in the article cited above), and (2) the product of a locally compact ANR with the Hilbert cube is a Hilbert cube manifold (by a result of Robert Edwards). Finally, it uses Chapman's results on the topological invariance of Whitehead torsion.

In any case, putting together the two results stated above, we conclude the following.

Any cell-like map between locally finite, countable CW-complexes is a simple homotopy equivalence.

To relate this to the actual question, first observe that simple homotopy type is not defined for every space. In the greatest generality I am aware of, it is definable for locally compact ANRs. Then being simple homotopy equivalent to a point is equivalent to being a compact contractible ANR. Since such a space is necessarily cell-like, we obtain the following answer to your last question.

Let $f:X\to Y$ be a proper map between locally finite, countable CW-complexes. Assume each fibre of $f$ is a contractible ANR, i.e. has trivial simple homotopy type (since the fibres are compact). Then $f$ is a cell-like map. By the above result, $f$ is a simple homotopy equivalence.

That is the most general statement I can find. If you only care about the case of finite CW-complexes, then the result admits a formulation with fewer classifiers, and was also proved a few years earlier. For example, it is stated explicitly as a theorem in page 17 of the article by Lacher cited below.

If $f:X\to Y$ is a cell-like map between compact ANRs, then $f$ is a simple homotopy equivalence. In particular, if $f:X\to Y$ is a map between finite CW-complexes whose fibres are all contractible ANRs, then $f$ is a simple homotopy equivalence.

Finally, two nice overviews of the theory of cell-like maps and related topics are:

Moreover, the first part of the article by Lacher contains an interesting exposition of theorems of Vietoris-Begle type, much like the ones mentioned in the question. It might be a good place to start looking for a general theory of such results.

cell-like maps. A cell-like map is one in which the fibres are cell-like spaces. These are compact spaces which are contractible in a very strong sense. Result 1: A cell-like map is exactly a map $f:X\to Y$ such that any restriction $f:f^{-1}(U)\to U$ is a proper homotopy equivalence for any $U$ open in $Y$. Result 2: even more interestingly, cell-like maps between several types of manifold-like spaces are approximable by (and homotopic to) homeomorphisms. See mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0111.0128.ocr.pdf for a nice overview. – Ricardo Andrade Mar 19 '13 at 7:36