OP, I'm a music teacher and I'm completely bemused by the responses you're getting. The questions you've asked should be no problem ( though I agree it's better to read up as much as possible on the teacher first, and ask questions in a phone call beforehand, not in front of the child. I agree also that tone and manner are important!)
But generally - I have to say that all those saying 'the school recommended her, of course she's qualified' are incredibly naïve. Likewise those saying 'you don't need a conservatoire-level training to teach young children'. In most other European countries it's the norm for practically all instrumental teachers to have that level of training, precisely because giving young beginners a good grounding is hard; it can cause massive problems later which can be hard to remedy if the child learns poor technique at the start; and it's generally possible to spot from a fairly early stage who's getting good teaching and who isn't- and I'm referring here to what you might call 'normal' children as opposed to the prodigy who practises a couple of hours per day from age 6.
There are a huge numbers of poorly-trained teachers in this country, because we have no expectation here of anything better - just look at the numbers of parents on this thread who seem to think it's a positive that they've no idea of their child's teacher's qualifications. Schools generally have no idea how to gauge whether an instrumental teacher can give a good grounding or not, so a recommendation from a school isn't necessarily worth what you'd think. Parents likewise often assume a child is making 'appropriate 'progress, when they might be being entirely inappropriately grounded. It's easy to think all this is unimportant, but poor teaching removes options from gifted children later on.
Also, poor teaching in generally still pretty expensive, and I wouldn't be willing to waste the money, myself. It's hard for parents in your position - in your place, I'd be able to get all the information I need via my network, and I have all sorts of means of gauging whether I'd regard someone as suitable to teach my own children (and I would be picky about this) but most parents don't have the background or contacts to make these type of judgments - but are still entitled to want to have their child appropriately taught.
Advice for the future: I'd tend to let this teacher go. She may be good, I don't know, but I'd be wary of a teacher who responds with 'I've been teaching for 30 years so I must know what I'm doing' - that's simply not true at all, and all good teachers should know this. Next time read up on the teacher first. What sort of training have they had? If they only have grade 8, they're not terribly advanced themselves, and chances are they don't have a sufficient level of technique to impart good basic technique to young beginners.
Don't bother asking about grades - for a start, it's not true that ABRSM is necessarily 'the best', and in any case, many really good teachers are not so keen on grades anyway. I'd be more impressed with a teacher who said, 'I don't tend to put them in for anything below grade 5 and I can prepare for whichever board suits her strengths best. But really, what I'm teaching is technique and performance skills, and if she stays, she'll get lots of performance opportunities and pick up grade 8 somewhere along the way'.
It should be fine to have this conversation (though I agree that tone, manner, and time and place are important, and it's possible you've got these slightly wrong!).
If you find someone who seems good and whose pupils do well, ask to be put on their waiting list.