Hi & welcome,
Unless you have other grazing animals for it to hang out with, I wouldn't suggest getting only one horse(whatever size). They do tend to survive, but they're herd animals & tend to stress & be unhappy alone. Like a dog. I like dogs & horses, so don't want to make them miserable, so wouldn't consider keeping a lone animal like that.
Also, while you do sound responsible, you don't mention a job or money to pay for stuff, and horses(of any size) are a lot more expensive to keep than guinea pigs. I suggest before you get your hopes up, even if your mum might help you pay, you work out your budget. There will be hay, supplements, farriery, dentistry, vet bills, worming and equipment to consider for a start.
Assuming you 'can do' and can get a couple of minis, or a mini & a goat or some such(does your mum knit?? Angora or sheep she can use wool from??
) then yes, a quarter acre will be adequate, and altho I agree with people's caution about too much feed for a mini, that area is likely to be eaten out very quickly, so you'll soon be left with a dirt yard & feeding hay full time(they need about 2% bodyweight daily in forage). As someone else suggested, you could always fence off half of it for them to live in, and let the rest grow & be grazed only part time.
Yes, poisonous plants need to be fenced off or got rid of, especially if there will be no grazing.