Shannon, Beegin and Brady (…Snow sucks.)
@hamtaro There's a few other suggestions above that could work - I also imagine that if you've researched how to get your dog to let you touch their paws with positive reinforcement that would be similar to picking up issues.
The biggest thing that I would add is that it doesn't have to be like a big thing. Add little steps here and there throughout your day with him. For example, when you hook him on his leash for a bathroom break, reach out and touch his side his belly whatever he's totally okay with first. Then slowly with each trip outside you move the touch until it is under him. You can reward periodically with treats but not all the time. Then you start with a little lift just enough to put pressure, keep doing this for a week etc. Then move to 2 hands then pick his front half a little off the ground and so on. It doesn't just have to be outside trips either. When you feed him, touch his belly. Whenever you pass him in the room, pet him and touch his belly/do mini lift etc. just so that there is just a lot of little neutral interactions in the direction you want to go throughout his daily life. Combine that with periodically having longer training sessions and you'll desensitize him to whatever was bothering him about being picked up. I hope that makes sense.
I would try a raincoat for the rain. It helped my guy to go out in the rain. He would just get really cold so the coat helped. I did the same thing another poster mentioned - bought a size larger and modified the belly strap to fit my B that way the coat covers his whole back.
My B will also do this - sometimes he's so set on not going somewhere he'll lay down. I usually give him a moment, then start back in the direction we're going with a really excited voice and a quick step like I'm about to run. I'll give our walking command "Let's go" and he thinks he might miss out on something exciting so he'll come with me.
If your guy nips/bites at being picked up, that's concerning. Firstly, make sure it's not a medical thing and check in with your vet - especially if it's been a while since he's gone to one. Then, you'll want to start positive reinforcement to change that behavior. Even if he doesn't get to the point of loving being picked up, he should tolerate it from you for safety sake.
I absolutely agree that crate training must be a positive and you have to get them to also feel safe in there. I like putting really high value items like a brand new bully stick or a PB or yogurt stuffed Kong in there with the door shut. Then let the b try to get it. Make him want to go in. Give him lots of repetitions of the command (I say "Go in Little House" but use whatever you decide) and differing lengths of time in the crate. I also always give some kind of a treat even if it's just one mini size one when he goes in even if I am letting him right back out again.
My guy does far less well in a crate in a car for traveling long distances so I also use a calming spray on the crate bedding and in the car. Comfort Zone (used to be called Adaptil I believe) works well for me.
I too had good luck with the yip strategy when raising my boy. To this day, even though he can be really grumpy when woken up (he's now going on 13) if his teeth touch any part of me accidentally or when playing with his stuffed toys, he freezes and stops. He never went so far as to give me face licks, but he would act as though he understood that he had hurt me. He would bow his head to approach me and/or lay down near me.
As a caveat though, my b has never really cared much for squeaky toys. He much prefers tug toys or the ones with crinkly plastic instead. He may have more readily given up on the "game" since he's not a fan a squeaky toys.
Yippie another potential basenji owner!
That being said kudos to you for doing the research on whether a b would fit your family! From my own experience owning a b, some of what a "basenji needs" can really vary with age. If you are looking at an older b, that has pros and cons. My own guy definitely slowed down around 8 to 9. Before that, 2 to 3 walks, toy time, training time or thinking toys were essential or he got into trouble. Luckily for me, he mostly just overturned laundry baskets and hid the items all over the house. Once older, he has been fine with 1 or 2 walks and a little play time/brain time. If I can do more, we do but he's a lot more patient. He has also never figured out chain link fence, but I have seen others who have. My guy figured out how to use the rotating rack inside an entertainment center cabinet to get out the squishy VCR boxes he liked to try to chew. They each learn nasty tricks in their own way.
That said, older dogs are a bit more set in their ways. I could never get him to not sleep with me at night now and he has never met a cat he couldn't chase. Took on a 35 lb Maine coon cat without batting an eye.
You'll definitely want to have some trial time in your home to see how he/she fits with the other animals. I would check on here and with AKC chapter near you for local breeders. They may have know of dogs retiring or BRAT fosters in the area plus breeders usually want the dog back if it doesn't work out. There's lots of really knowledge breeders and BRAT folks on this forum - just ask!