And for your other points:
The docs for a basic roll-over seem to be lacking a bit around how the index write alias is managed.
Does it have to exist on the index or will ISM add it?
The write alias is something you have to kickstart on the first index you want to have rollover. After that the rollover request will change the is_write_index settings for you when rolling over the index. This is only the case if the alias is pointing to multiple indices. If you have your alias only on your single index then you don’t have to worry about the is_write_index at all.
When an index “Rolls over” for various conditions, does the new index automatically get the alias and the previous index has the alias removed?
If the alias is pointing to a single index then yes, the rollover request will create the new index, add the alias to the new index, and then remove it from the old index.
If the alias is pointing to multiple indices then it will create the new index and use the is_write_index to control which one docs are added to.
Not sure if it was my setup or what but I just tested a daily roll over and I had 5 indices all with the same alias (the one I use to write)
It depends on your original setup. You can read more about rollover here.