Once you out-trade a villager's specific option, you lose it: it has a large red X on it, and you cannot make that particular trade anymore. Is there any way to recover options that fall to this fate?
Yes, you need to trade with them using one of their other trade options.
From the Minecraft Wiki (emphasis mine):
Villagers will deactivate an offer if the offer has been used some number of times and it is not the villager's only offer. The chance of an offer's deactivation is random, but an offer must be used at least 3 times before it is eligible for deactivation. After an offer has been used 13 times, it is guaranteed to be deactivated if you close the interface after trading it (this is true even if all 13 trades have been done at once). Trading the last offer may activate an offer again.
Yes, you should be able to use that offer again if you simply preform some other trades with that Villager specifically. To avoid the trade being blocked, simple make sure to not trade over 7 or so times at once and wait a while between trade sessions.
When a villager goes to its trading station, for example a tool smith going to a smithing table, the villager replenishes his trades because he replenishes his stock.
No, the specific trade that has an X on it is not able to be done again. The villager will have other trades, but that specific trade is unavailable forever.