I'm wondering if there are consistent (or at least fairly reliable) rules for how to read numbers next to irregular counters, i.e. counters where you can't just put together the number and counter word in every situation. For example, the counter さい, the first number is not いちさい but いっさい. This is as opposed to counters like まい, where all you need to do to form the counter is add the number (e.g. ろく) and the counter, e.g. ろくまい.
The changes you're talking about are actually pretty regular. There are some irregular ones (一人【ひとり】、二人【ふたり】、三階【さんがい】), but what you're looking for is pretty straight-forward.
Wikipedia has a pretty good summary of the sound changes.