What's the easiest way to check if a slave at the auction is one we've owned before?
Also, does selling a slave cause a drop in their loyalty/obedience or an increase in fear?
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There are no in-game methods to be certain if a slave at the auction is one that you've owned before. The best you could do in-game is to use mind reading to check if their courage, confidence, wit, and charm are extremely high or low values as slaves are generated with fairly average values. You could check their stats in the raw save file as there are several stats like stress, lust, lewdness, which will all be zero for newly generated auction slaves.
No, selling a slave has no impact on the stats of the slave, which is why some strategies utilize selling slaves as a somewhat expensive mechanic to get around certain problems. In fact, slaves that have been sold effectively enter a form of stasis(assuming they don't get deleted) as only slaves in the player's slave list experience end of day updates, so any slave you sell will be bought back in exactly the same condition as you sold it.
So... does "if slave.[variable] > 0" work for any specific reliable variable? There's a variable for days spent in my household, right? That one would be 0 for newly generated slaves or ones I caught that day, but higher than 0 for slaves I've owned longer than a day?
What I'm basically after is switch the slave's reaction to me perusing them at the auction, based on whether they're familiar with me. So I don't care about first-day slaves at all, but about slaves I sold after molding them a bit, basically. (I routinely sell and rebuy a slave to get upgrade points and a free(ish) brand. I don't think a slave like that should react to my presence the same as a slave who's never been in my house.)
Also hang on, slaves at the auction house have a stress level of 0? That seems... interesting, from a worldbuilding viewpoint. I might toy with that one.
Every variable has it's own set of cases where it can be less reliable, so using any single variable will be prone to problems. For instance, "person.metrics.ownership" would be the variable that most consistently corresponds to days spent in your household, except that it actually counts days in your slave list. This means that time away for quests, vacations, training, or recovery still counts towards days of ownership.
Newly generated slaves don't really have any backstory and stores and slave auctions don't particularly constrain themselves to the rules of the world. Though only a few versions ago this was basically true for any time a slave was away from the mansion. Even now you can indefinitely lengthen the duration of a pregnancy by simply using the Dream spell on the slave every night.
Trying to make sure I don't misunderstand this: If your slave goes away from the household for some reason, but has not been sold etc., it still adds 1 to the Ownership counter for each day? So like, a person who's been working in your mansion for a week, and a person who spent 6 of the 7 days out of the rotation due to vacation/recovery/whatever, would have identical numbers added to their variable?
That feels like it's working as it should be. Feels like if the number of days owned were calculated minus "days outside the rotation", it'd wind up with way more problems.