Do I ever run out of new recruits if too many die, or do I have an endless supply of cannon fodder to replace their fallen comrades?
You will run out of rookies in your base, however you can hire new ones that take a few days to arrive.
As long as you have the money and the space in your barracks, you should be able to recruit new rookies. With the "New Guy" upgrade from the Officer Training School, the new recruits can arrive as squaddies rather than rookie.
You will eventually have to hire new recruits at a low price:
XCOM starts out with a group of twelve Rookie soldiers. Additional soldiers can be recruited via the Barracks and arrive at headquarters three days later. Each recruit costs §10 on Easy and Normal difficulties, and §15 on Classic and Impossible. Additional soldiers can also be received as a mission reward from the Council.