For the first four questions, are you just asking what is the maximal dimension of a fiber of the blow-up map : $p : \mathrm{Bl}_{Y}(X) \longrightarrow X$? If that is the case, then for $X$ and $Y$ integral, the maximum of dimension of a fiber is $\dim X -1$ and it is attained when you blow-up a point in $X$.

The last question can be made into an interesting question if you transform it slightly. The dimension of the fibers can be arbitrary between $0$ and $d-1$, whatever the type of singularities, so nothing's interesting there. On the other hand, you could ask what is the relation between the **variation** of dimensions of the fibers of the blow-up along $Y$ and the **variation** of singularities of $X$ along $Y$.

This topic was studied a lot in the 60's, mostly in connection with Hironaka resolution's of singularities. Namely, Hironoka says that $X$ is *normally flat* along $Y$ if the exceptionnal divisor of the blow-up of $X$ along $Y$ is flat over $Y$. In particular, all fibers of the blow-up have the same dimension over points of $Y$. In case $Y$ is smooth, normal flatness is a notion of equi-singularity in a very strong sense. You can have a look at [this question][1]. It implies that all points of $Y$ have the same multiplicity in $X$, but in general it implies much more : namely that all generic polars varieties with respect to $X$ which contain $Y$ have the same multiplicity along $Y$ (this is a [hard Theorem of Teissier][2]). 

Let me restrict to a simple case where things can be easily can be computed : assume that $X$ is a divisor in a smooth variety. If $Y$ is a smooth subvariety of $X$, then $X$ is **normally flat** along $Y$ if and only if all points of $Y$ have the **same multiplicity** in $X$. This computation in the case of hypersurfaces in ambient smooth varieties has been first carried out by Hironaka himself, but I think that Lejeune-Jalabert and Teissier gave a [simpler approach][3] to this result.

Very recently, this notion of normal flatness has proved to be quite useful in the context of non-commutative resolutions of singularities.


EDIT : The answer to question 3 is no. Indeed let $\mathcal{N}_n$ be the nilpotent cone for $\mathrm{SL}_n$. The springer resolution:
$$ \mu :  T^* \left(\mathrm{SL_n}/B \right) \longrightarrow \mathcal{N}_n$$ is birational (a resolution of singularities) and for any $x \in \mathcal{O}_{subreg}$ , we have $\dim \mu^{-1}(x) \geq 1$ (where $\mathcal{O}_{subreg}$ is the subregular orbit). Since $\mathrm{codim}(\mathcal{O}_{subreg} \subset \mathcal{N}_n) = 2$, we deduce that the Richardson map is the blow-up along a subvariety of codimension $2$.

Finally the fiber of $\mu$ over $0$ is the complete flag varieties of $\mathrm{SL}_n$, it has dimension $\frac{n(n-1)}{2}$.


  [1]: https://mathoverflow.net/questions/263752/whitney-conditions-vs-equisingularity/263758#263758
  [2]: https://webusers.imj-prg.fr/~bernard.teissier/documents/Cargese.pdf
  [3]: https://hal.archives-ouvertes.fr/hal-01053223