Jammie I'd echo the advice to get her teeth checked. If her breath is bad then she could have an infection, and if they are left untreated they can cause serious heart problems. It may be that she needs a dental (clean under anaesthetic).
Batshit it is completely dependent on the dog. Most greys love children, but whether they'd be suited to tinies depends on their personality. For example, my boy would adore small kids but wouldn't be suitable because he's large and bouncy and would be likely to knock them over and injure them - completely without meaning it. My old boy was perfect with small kids because he was very steady and patient and loved nothing more than being fussed over for hours on end. A good rescue will know which greys are suitable for young kids. However any dog will require supervision with tinies no matter how good they are. And it's vital that the dog has somewhere quiet and private to retreat if they feel they've had enough.
Xmas again it depends on the dog. Very few "need" a run, but most of them enjoy it. However if you don't have anywhere secure to let them off then it's not essential - regular short walks would be more important. They don't need much exercise. My girl is happy with going out just long enough to do her business and then come straight back in again - she doesn't like going outside of the garden as she's scared and is not bothered about being out at all. My boy loves being outside and running about but he only needs about 10 minutes before he's had enough.
The property we lived in prior to this had no garden so they were always walked on lead and were fine (and the rescue knew this). The re-homing advice will always be that you must not let them off unless you are somewhere enclosed - and because they can jump then the enclosure fencing needs to be at least 6 feet high. Lots of rescues - and also riding schools - have secure paddocks that you can hire for 30-60 mins if you want somewhere safe to let them off.
Mard we used to call my old boy "Mr Castanets" as his teeth used to chatter and his chin would wobble madly whenever he was happy or excited. The other two also "chatter" if they are particularly chirpy.
I'd also agree with Jumpers about the "house sense". My old boy had been in a home before and was very well trained and never had an accident in the house until last year (which was age related). My girl had a couple of accidents but was clean after a few days and was also well behaved. She is an ex-racer but had been fostered before adoption so had been house-trained. My new boy has never raced but also never been a house dog so needed training from scratch. He's been here 2 months now and he's pretty consistently clean now, chewing is becoming less of an issue and he's settling down nicely. The hardest thing was teaching him to walk up stairs because he's so big and heavy. It took him a week and he now likes to show off by walking up and down them backwards (he is a massive idiot!).