I've liveried at a large riding school for years and am very happy although it probably wouldn't suit everyone - the pluses are great facilities, easy access to lessons and clinics, competitions on site, lots of people around all the time so very social and always someone available to lend a hand and give unasked for advice or as a hacking or competition buddy. The minuses are that it is relatively expensive, busy and noisy at all times of day (great for de-spooking your horse, not so great if you have a stressy one!), loads of not-always-well-trained children and dogs underfoot, can be cliquey and gossipy, your stuff is never safe unless under lock and key. School access is a perennial issue, we have 2 very large arenas plus 2 smaller ones that can be used for lunging etc, but you'd struggle to find somewhere to ride in the weekend day time (pre corona of course!) as there will be anywhere up to 7 or 8 lessons going on at once and they take priority. Not a problem for me as I ride at less busy times or hack out anyway but would get very annoying if that's your only real riding time of the week.
Questions I'd ask would be the same as at any livery yard, e.g. what are the yard 'rules', 'opening hours', security measures, does anyone live on site, emergency procedures, are children allowed unsupervised on the yard and if so from what age, what facilities they have (jumps, hacking etc) and any restrictions on using them. Does the farrier, dentist, physio visit on a regular schedule and what are payment arrangements for that. What storage do you get as a livery and is it secure (e.g. space in locked tack room, lockers etc).
I'd want to know what is included on their full livery package (or part or working livery if that's your plan) and what do you have to pay extra for, common 'extras' include riding or exercising pony if you can't make it to the yard, tack cleaning, bringing in and holding pony for vet/farrier/dentist, medicines and supplements, tack cleaning, clipping, grooming. Usually everything else is included ie basic feed, bedding, hay, mucking out, bringing in/turning out, care and maintenance of the fields - but worth checking so you can be sure you are comparing like with like price-wise.
My number 1 question for any yard would be about turnout, how much can they guarantee especially over winter as this is so important for horses' wellbeing, 24/7/365 is the gold standard but a lot of yards will insist ponies are brought in overnight in winter to save the fields from poaching. I'd want to know if there is any reduction in livery charges if pony does live out 24/7 all year round. The 2 field accidents could just be bad luck but I'd want to know what their procedures are for managing herd dynamics and also I'd want to see that the fields are well maintained, fences and gates are in good repair and any hazards e.g. ditches fenced off.
In general (and I know it's hard to tell) I'd try and go with the yard that gives you the best vibe, not necessarily the biggest, smartest or cheapest. As you are not horsey yourself you are going to be placing a lot of trust and giving a lot of money to the yard you choose and you want them to be friendly, knowledgeable, caring people who are going to alert you to any issues asap and welcome you and your DD coming to them with questions and for help, not the sort who treat clients as a nuisance who should be seen-and-not-heard or worse just sources of £££. You really want the yard to be a happy place for you and DD as horse ownership comes with enough hassle and stress as it is without adding any unhelpful yard dynamics into the mix!