Hey there! Congrats on your new bun bun (photos are welcome! )
First questions first, how old is your bun and is she spayed already? If she’s not spayed, this is where ALL your problems lie and you should get her spayed as quickly as possible so she doesn’t pick up any bad habits. Typically around 4-6 months she can safely be spayed but you’ll want to find a rabbit-savvy vet who knows what they are doing. If your vet thinks your rabbit needs to fast before their spay (red flags should go up in your head! And I would look for a new one.) The House Rabbit Society has a whole article on how to find and questions to ask about finding a good vet so I would check out that website to get some incite into finding a good one.
In my opinion, you can NEVER have a rabbit out too long. Especially if they are young. Which from everything you have described your bunny really does sound young! (I know, I had three of the little rascals myself). If she’s young then a lot of that energy will typically calm down once she is spayed and a few months older. (9 months of age they really tend to start calming down and not need as much out time). ALTHOUGH, if you’re having trouble with litter box habits I would suggest limiting her space as it’s MUCH easier to teach litter box habits in a small space and make it bigger as she begins to get the hang of it. Usually as I expand an area, I add a litter box so they realize it’s there and as she gets better you can take out the ones she doesn’t use or rarely uses.
Lots of young bun buns take a few months to really get into the hang of veggies. I had to poke and prod him (literally) with a stalk of cilantro for about a month until he got so annoyed he grabbed it out of my hand to throw it. He must’ve gotten a taste because all of sudden he was a fanatic for cilantro. Carrots are bad for rabbits so when starting out with veggies definitely do NOT start out with carrots. Carrots should be given in VERY small amounts and my buns won’t even touch carrot so I wouldn’t be too surprised if yours isnt either. Try something very fragrant like basil, cilantro, or another herb. She’ll be more tempted to try a bite. As long as she’s getting the right amount of pellets though, her choosing not to eat veggies isn’t a big deal although I suspect the issue is she just hasn’t been introduced. If she hasn’t, then you should only introduce one veggie at a time (about one a week) so you can monitor her poos to make sure none of the veggies give her runny or loose poos. If so, then I would lay off that particular veggie and try different ones until you figure out which ones work well with your bun and her tummy.
Hope this helped! And welcome to BB!