Hi and welcome to the world of owning mini's and horses
how many horses do you have? and which part of the world are you from
ok I'm not sure if your mare is pregnant or not, but I had a similar case with one of my girls Penny, she started getting a tummy and a small udder and I could get milk from it... she was bigger then your girl and we all thought she was pregnant. I got her a year ago in march and dont have a stallion so I knew she had to be coming up to full term... any way the 1 year came and wwent and nothing... she has had foals before and some mares will keep a small amount of udder and milk after they have had foals even though Penny hadn't had a foal in 4 years!
has your mare had foals before do you know? I suppose with getting them at auction its hard to know... but do you have any clue if she was with a stallion? here are some pics of Penny so you can see.... she looked quite similar to your girl I think...
look at that tummy gotta be preggers right? wrong... lol
udder
a similar view to your pic.
I'm not saying that she isn't pregnant... but I'm not saying she is... it would be a total guessing game for you if she is especially if you have no way of contacting the old owners... keep a really close eye on her... you will see changes if she is...
also the change in attitude (I'm no expert...) but maybe she is in heat? if you have had her a month... this could be a possibility...
with regards to your other questions... I would take the halters off them while you aren't around... to risky to keep them on,... so many accidents happen with halters being left on horses... I don't want to scare you but just so you are aware.
you don't have to clip them unless its really really hot and they are suffering from it. if you are coming into Spring spend as much time as possible out there currying there coat. help them shed it. as you haven't had them for very long I would build up there trust before possibly scaring them with the clipper blades if you understand what I'm saying
with hay, if your horses are out on grass, then they might only need a biscuit or half a biscuit (flake ) a day or maybe even less, depending on their condition. If they aren't on grass then it is best to spread the hay out so they can still forage (which is very good for them) and it takes longer for them to eat it
how much hard feed are you giving them? are they all mini's? what are you feeding them?
we are here to help and I hope I don't sound harsh with anything I am saying...
would love to see some more pics of your girl please and will be able to help more then.
the other ladies on this site are brilliant and have heaps more experience and wisdom then me, they are all sleeping at the moment hehe (well should be anyways LOL) but hopefully they will come on here and share some of their knowledge.
Welcome again, hope you stay around
Cassie
...my kids were washing her and found out she has milk coming out...not dripping but when u squeeze her tits it shoots milk out...so my ? is do u think she is pregnant from pics and what are the signs of pregnancy? 
