hullwhite - Hull White Tree Calibration 2

This is actually to extend the question I asked previously and to follow up Bernd's answers. This is the original link: Instruments for calibrating Hull White Model

1.

As Bernd mentioned, it's generally a good idea to price a products using Curves/Models that are calibrated from quotes of the same (or similar) kind of products. My real scenario is to price hundreds and thousands of bermuda callable bonds. Does that mean ideally I have to calibrate trees for all of them based on their market prices?

2.

The main goal is to solve OASs, given it is an unknown variable, how can I calibrate the tree to derive mean reversion and volatilities? Mathematically seems to be unfeasible. As I have to optimize three variables - OAS, alpha and sigma.

3.

Another further question I have is assuming I am calibrating a quarterly tree for 30 years use swaptions. The calibration process is essentially solving forward volatilities for every path iteratively, right? What if there is no swaption from the market that exactly matches one of the paths, how can I derive that forward volatility? For example, I want to solve the volatility for the path of 29th year to 29th year and a quarter, but there is no such swaption in the market, how can I calibrate that part of the tree?

4.

Last question is if I want to calibrate a tree for 30 years, what's the ideal path number should be used? I am right now using quarterly path that is totally 120 paths.

Thanks!