I just wondered how the bridge pins could have been pushed in "too far", as the string ends themselves will form a natural barrier to them being too far in.

Pliers are not a good option because, as you've found out, the pins can get broken - I've done it myself. A better option is to use a tuner roller of the sort which has a slot in the handle. The slot fits under the head of the pin, which can then be levered up gently.

As to pin sizes, there is no standard, and there are variations in width, so you may have to take it in to a shop and see what fits practically.