Hi and welcome.
First, where are you located? It's always good to have local support! Also, which rescue group? What do you mean by "dog farm"? Do we get to see pictures? We love pictures. :)
LOL on the panic. You are far better than I was at this point. I had owned Chows and chow mixes most of my life and then in early 30s (age not 1930s :) ) Rottweilers. I wanted a small dog that didn't need grooming (I really believe grooming the chows is why I have carpal tunnel), not yappy, and something my daughter could show. Oh my word. So the breeder tells me that her friends in the basenji world are horrified she would let someone with a chow and Rottweilers have one of her dogs. She also didn't want to sell me Sayblee because she said she was too strong and hard of a dog. I actually laughed at her. I reminded her that your dog will be about 25 pounds... that I did training and aggression rehab with Rottweilers, Chows etc. How hard could a 25 pound dog be.
I know before bed every night she laughs. I got Sayblee in 2001. I have never loved a dog as much in my life, and never had anything so utterly animal aggressive. She finished her championship and I spayed her. I got a lot of flack, but her aggression didn't need to be passed on. I lost her to lymphoma at 7 yrs old. I still cry when I type her name or look at her pictures. (We got her double niece 2 yrs later and have a wonderful rescue from Camp Basenji (it was originally BRAT, but Pam created her own rescue later).
So what do you expect? You won't know until she arrives. When you mix any breed, characteristics might or might not be one or the other or even a balanced mix. If the basenji shines through, expect eye contact to be something she likes. Unlike so many breeds who react to looking into their eyes as a challenge, they'll look back, hopefully with love and not with the "you're not the boss of me" look! You can expect to have your heart stolen no matter what she is. As for sleeping in the bed... lol that's preferred, but I can tell you my daughter will never have another in her bed after Cara. So it isn't the only place. However, let us know how long before one or the other kid slip her into their bed if you don't. :)
Most rescues won't place most dogs with only invisible fencing. I absolutely would not ever. Instead of me going into it.. call about 5 shelters around you and ask what they think of them. They'll tell you how often dogs come in with electric fence collars on. All breeds, not just basenjis, can get excited and run through. Problem is, once out, not likely they'll run back through it. Also, your invisible fence provides NO protection for your dog... other dogs, a variety of larger wildlife that are indigenous to your area, people... walk right in. Ditto on any kind of line tying them out (not to mention basenjis do seem to have razor blades they pop out only for cutting leashes and stuff). If you absolutely cannot put up a fence, then you use your human family resources and take the dog out to potty, out to play, and on walks. If the fence is in place and you have someone out there all the time with her, then if she does charge through, you know hopefully before she gets hit by a car or attacked by another dog.
As for clutter... not going to lie. Basenjis are the best house keeping trainers around. Your children will learn to put up their toys and easily destroyed items... or put in their bedroom and close the doors.