There are different ways of doing it. A very common way is to download a list of all available WIS players for any given season, and then to delete all players who are listed on keeper rosters. This then gives you your available draft pool.
As to your second question, in most historical seasons the answer is no. I would hazard that only from 1993 on (when MLB expanded to 28 teams) are there enough players to staff a team, and even then there are no guarantees.
If you really want to use older seasons without including AAA, you probably have to consider either smaller progressives (eg 12-16 teams) and/or run a multi-season progressive (for example, use players from 1901 and 1950 in season 1, then 1902 and 1951 in season 2, etc.)