If a user is new and receives down votes, they cannot get negative reputation (or even non-positive, i.e., 0) right? So if that user gets down votes, then earns reputation, does the loss of rep they would have earned from the down votes continue, effectively preventing them from earning rep until they have made up for those down votes, or is it simply lost when they are at 1?
Votes are always applied in order. So any down-vote after the user hits 1 has no effect on the users reputation.
If a new user gets 5 down-votes followed by 1 up vote they'll have 11 points. This will be true even after a recalculation.
Retracted due to inaccuracy; see comments
My understanding is that their normalised rep will not show any effect (as you say, that cannot drop below 1) but that it will effect their actual reputation. That is to say until their account's reputation is recalculated the effect will not be seen on the site.
So if someone on 1 rep has 5 downvotes they will be on 1. An upvote will up them on 11. After a rep-recalc they will really be on 6.
If you're not aware your actual rep can be see at https://superuser.com/reputation, although you can only see your own not others.
Some events (such as I describe here, but also deleted or migrated questions/answers among other things) cause the normalised rep stored and shown on the site to deviate from the actual value for the user.