Hi kathryn, that playhouse looks like a great place for a bun! As long as it can be locked securely, and has enough ventilation, I wouldn't worry about her being outside in it here in California, at least not in the winter. If you provide plenty of hay, hiding boxes and toys, your bun will be fine. The summer will be harder, though... I don't know how you could cool it enough.
There have actually been a few threads here on the forum about housing buns outside, and what hutches to get for them.
A lot of people in the UK for instance house their buns outside.
You could iook at these threads for ideas:
http://www.rabbitsonline.net/view_topic.php?id=27041
http://www.rabbitsonline.net/view_topic.php?id=8348&forum_id=1
There are more, including one with lots of pictures but I can't seem to find it right now.
For now, to keep her from soiling the carpet in your room while she's out of her cage, you could do as someone already suggested: get some plastic table cloths from the dollar store and put those under her run, with some scraps of carpet or an old blanket on top for her comfort (That's what i do in the areas where ours have their "poop and pee wars" outside the gates between their areas). That way, she won't be on the carpet, and atleast your dad can't complain about that anymore
. This will also work under her cage, and if you extend it past the sides, anything that does fall out will be atleast contained off the carpet.
The only problem I can see with housing Bunny outside is your lack of time. Outside buns take a lot of work, and if you are always as tired as you are now, I can't see how you could manage to also go out and feed, clean and socialize with Bunny. It doesn't look like you will get any support from your family, either.
Could you get your grandma who gave you the rabbit to talk to your parents and explain to them about rabbits as pets? I hope she atleast knows about them, though of course not talking to your parents before giving you one looks like the wrong thing to do.
I hope you can make things work, but if you can't get any support right now, you may well have to choose between your sports or your rabbit. I know that is a cruel thing to say, and a very hard decision, but it looks like you already have not enough time and energy to treat Bunny the way she deserves, and no support and help from family members either.
What are you going to do if Bunny needs to see a vet, needs long term care? Can your Grandma help in any way, housing Bunny for you for a while untill she is spayed and her hormones (and your family) calm down? Could she take care of some of the vet bills for you?
I do hope you can eventually talk your family into letting you keep Bunny, but a lot will depend on how well you can take care of her, possibly with help from Grandma.