I'm no expert but all this sounds like normal Basenji behavior especially for an under-exercised, under-stimulated and YOUNG dog as already noted here. This is not to say your dog isn't getting enough exercise to stay physically healthy, but pretty much every dog will be better behaved the more it is getting tuckered out.
I have raised two Basenji puppies recently, now 1 and 2 y/o, and both are still pretty rambunctious (typical young Basenjis) especially in the winter when it is no doubt tough on them and my girlfriend and I to want to get outside. We live in an apartment but the building was once a hotel so we have interior hallways that take us about 10 minutes to walk all three floors. I know we have a peculiar living situation but you might be able to find something similar (just indoor hallways or maybe outdoors under the cover of some structure) Also, just because your dog doesn't want to go outside because of cold or rain or snow doesn't necessarily mean they shouldn't. Your judgment of what they can take is more realistic and optimistic than their own. Boots and jackets can do wonders if the temperatures necessitate it but either way, if it's not uncomfortably cold out just get out there! If I only walked when my Basenjis wanted to in the winter, we'd walk once a day and I'd be cleaning up my floors twice a day!! They are definitely not cold weather dogs but they can hesitate unnecessarily even when it's not cold enough to need boots and sweaters. Just exercise sound judgment: keep an eye on the temperature outside, feel her paws and body on walks, notice if she's shivering or lifting her paws.
Even with more outdoor exercise, more positive and engaging stimulation at home is definitely called for. Everything she's doing now that you don't like is her making up her own fun. As already mentioned, a fun and yummy training session can not only be used to teach desired behaviors but will also tire a dog out. Why doesn't your get social time with other dogs at day care? If it's because of aggression, definitely read up and/or get training to mitigate that. If your dog could socialize with other dogs at day care, she would get tired from running around AND using her brain to socialize (the latter of which is not readily obtained from regular walks). Then you'd be getting your money's worth for the day care and for a happy (tired) dog. If no socialization is just an attribute of the day care itself, consider finding a day care that does include that.
Speaking of socialization, you should try to find a fully-fenced dog park to visit (assuming she doesn't have major aggression issues) to get the same exercise and socialization benefits of day care in a likely larger, more natural space full of sights and smells. As already noted here, any time a dog has to use their brains for more than just walking around, they get tired at least twice as fast. When I take my dogs to the dog park and there's lots of dogs there, of course they run like crazy but they do so much more than that with the other dogs, trying to figure them out, how to get them to chase or get them to run so they can give chase, how to sniff the butt of a Great Dane, what all that hair is for on an English Sheepdog. A properly socialized dog that "explores" and plays with other dogs for an hour will pass out on the couch like you just walked them for 3 hours. If my dogs weren't well socialized and we couldn't go to dog parks at all, we would all be going insane!! And when we go to the dog park and there's not a lot of other dogs there my dogs and I get bummed. Being off-leash is great, but it's the other dogs that make the experience so engaging and thus tiring! I think the only thing that would be more engaging would be hunting with other dogs. Note that part of my dogs socialization was going to dog park from 6 months of age and being very attentive to what they were experiencing…not letting them get run over by bigger dogs, avoiding aggressive dogs, etc., we didn't want to get off on the wrong foot with such a valuable socialization and exercise resource. With that cautious start, now they're some of the most rough and tumble dogs there (in a good way) even though they are usually the smallest playful dogs present and they LOVE strangers...they are overflowing with healthy confidence. [Funny story: I had a guy ask me "those aren't Basenjis are they!?" I said yes. Then he said "that's odd, I thought Basenjis were very aloof, unfriendly, and aggressive with strangers and other dogs". Needless to say my dogs practically licking his face corrected his Basenji misconceptions which probably originated from under/unsocialized Basenjis which unfortunately are fairly common.]
Once your dog is getting worn out every so often, her destructive and aggressive behaviors will likely improve. Revisit all of the training you two have already been through (don't use it, you and her will lose it). Get your family involved. Include training to specifically address the behaviors you don't like and to get her to do things you do like instead. EX: give her substitutes to bad behaviors; try not to put her situations where you know she's likely to do what you don't like. I have yet to meet a dog that preferred biting people for fun over chewing on a fresh bone (which is great mental stimulation, too, like a Rubix cube for dogs). If she likes to bite you in specific situations, try to avoid them or at least don't allow her to get latched on (saying "no" when she's already pulling on your skin is a little late, she's already got what she wants). If "come"/recall isn't working, work on other commands for some time then come back to recall. Definitely get all the books agilebasenji mentioned…anyone who can get a Basenji to do more than 3-4 commands knows what they're talking about and that's no joke!!! It's pretty well known about the breed that when they're young, if they're out of their crate you either have to "puppy-proof" their entire run-free area OR constantly watch them. They are a mischievous, often destructive breed especially as puppies. Practically every one that has raised a Basenji puppy has ended up having to do things that would make other dogs owners cringe thinking about the effort required. I often describe Basenjis as the "anti-Lab". Plan accordingly.
can help keep her mind busy and energy burned off some. And if the daycare doesn't let her play with other dogs, what are you paying for? Do they walk her a LOT, play with her one-on-one? If not, might be no more expensive and more beneficial to have someone come over once a day mid day for an hour walk.
