Looks and sounds lovely but as you say maybe even too good to be true so do ask lots of questions! I think in terms of questions to ask, lots of probing about the terms of the 'deal' i.e. when she says just carrots and grooming I would assume from that (a) she doesn't want any kind of financial contribution at all and (b) horse is kept on full livery so all care and chores are taken care of, so you would effectively just turn up and ride but you absolutely need to clarify this!
Also even within 'just turn up and ride' there are differences in reality, at my yard even for a full livery so no yardwork to do, just turning up for a ride can mean trekking to the furthest corner of the far field if you want to ride at a time the herd is out - a 10-15 minute walk there and same again back (that's assuming horse is good to catch and you don't also have to chase them around for 15 minutes waving polos like my little 'darling' 
). Then rugs off, wash down muddy sticky legs, hose down the wash area, groom, tack up etc. Then afterwards untack, brush off, wash legs again (our school is semi flooded and horribly mucky right now), wipe down tack, rugs back on - some horses would have a feed before going back out so wait for them to eat that, skip out stable/tidy any piles - then walk back out to far field again - probably 1.5 hours' worth of jobs not even accounting for riding time, a bit different to if you are used to simply turning up for a lesson and horse being ready and waiting for you! Obviously that's a bit of a worst case scenario but if she actually wants you to occasionally muck out or turn beds down or pick up a barrow load of poo or other minor jobs (things as a horse owner you do so much as a matter of course you don't even count them in your head!) it might turn out to be quite a time commitment so you do need to be honest with yourself and her about how much you can commit to especially in the very worst of the winter weather/dark!
Also of course ask about the horse, how is he to ride, does anything spook him, what is his 'go-to' if naughty, upset or scared - if she says 'nothing' I'd be a bit suspicious because even the most bombproof horse does something wrong sometimes, even just stopping dead still and refusing to move! That's ideal but if he bucks, spins, naps etc even if very rarely I'd want to know so you can be prepared and know what to do. Ask about his routine, winter and summer, and also her's, does she stick to a very strong routine or is she often away/busy and might want you to cover extra days here and there, what about holidays etc? Will it be a problem if you ever need to not cover a day due to sickness or childcare etc? Has the horse ever been lame or sick, what are the signs to look out for to see if anything isn't right with him? Who if anyone on the yard will be around when you are riding and who can keep an eye on you/you can ask for help? What are the yard 'rules' re DC, opening/closing times, security, covid measures, school use (who has priority etc), hacking out? Can you have lessons on the horse (I'd really really recommend this)?
Re the etiquette when you go to see him, I would expect her to go about her normal business and you to 'shadow', i.e. bring him in if he's out in the field, groom, tack up together - this gives you a chance to ask your questions about his routine, his tack and how to check it's fitting right etc. Then for her to ride first and you to watch - do feel free to walk away if your gut is telling you something isn't right (e.g. she makes some excuse to not ride herself and wants you to ride first, or horse is playing up), never ever get on a horse you don't feel safe/happy on without an instructor or other trusted person there - but if you and she are both comfortable you can then have a short ride - probably a walk, trot, canter on both reins and maybe a little pootle out up the lane/road with her walking beside you. I know it's nerve wracking but I would try hard to just ride as you normally would in your lessons, both you and she want a realistic picture of if the horse is suitable and as I say if your gut is telling you no then the sensible thing to do is walk away... good luck!