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Her face looks Basenji, maybe not the muzzle however. Are they destructive? Can be, don't have to be. I don't like to crate and I never have, beyond the initial puppy stage or with a rescue until I can get a handle on what to expect. I do restrict access to areas of the house that might be vulnerable until I am sure of the dog. Not all can be trusted to have free rein. Be consistent in what you expect or will tolerate, and be very watchful until you understand what behaviour is "normal" for your new dog.
Are they really that active? I would say no. She is likely excited to be in a new place and curious about everything. She should settle down. Of course, with a rescue there may well be reasons someone gave up on her, and it's not always immediately apparent if a dog has problems. Hopefully she will fit in well with your family.
Invisible fence......likely not a good choice for a Basenji, particularly if there are coyotes in your area. You need to protect her. Coyotes will go after a small dog, and sometimes even a not so small one. I know someone who lost a Jack Russell when he was lured by coyotes and then attacked once he was beyond his master's protection.....and Jack Russells are tough little dogs! However, invisible fence can be useful to back up a physical fence if your dog turns out to be an escape artist. Regarding landscaping.....there are some deer fences that blend well and become almost invisible. Perhaps combined with invisible fence that might be a good option. Otherwise, walks or biking your dog should help, and definitely challenge her mentally. Basenjis are clever and I believe they enjoy learning. I taught my boy to retrieve objects by name.....it was fun for both of us.
Good luck with her.
