This is tricky and you will need to be paying attention. Dogs are very fertile and fertile matings can occur five days before and six days after ovulation. So you see you're dealing with an extended time. The good news is that it's more like 11 rather than 21 days. The bad news is that unless you are hormone testing, you won't know exactly where those 11 days fall.
FWIW I think the manual says something like "generally speaking, Basenjis in the Northern Hemisphere come into season about late fall/early winter". She's likely relying on those two little "wiggle words"!
Good luck. I'm sure you'll find the whole experience interesting if a bit of a PITA.
I'm sorry this happened. I'll pile on and say that dog parks are not a place for toys or treats, with the possible exception of balls that are being thrown and retrieved. I don't think I've ever known anyone who went to a dog park regularly who didn't have or witness a problem encounter.
That said, I've seen people walk into dog parks with hamburgers.
The way to avoid this is to avoid dog parks where this happens. If possible try and find one where the people aren't clueless -- which may be difficult or impossible. If you have to go, try and go at off hours when traffic is light. Problems usually show up where there are more than just a few dogs.
As far as playmates are concerned, yes Basenjis will be better off with larger dogs. They are rough and tough and usually too fast and strong for dogs of their size or smaller, unless those dogs are also bred for hunting.
I think you need to look at your dog to figure this out. Some seem to mature faster than others. I think I waited until our youngest girl was 18 months before starting on more strenuous activities.
That said, I don't consider walking or running to be strenuous. I'd reserve that for activities like jumping where more stress is applied to the joints. For hiking, I'd be more concerned about paws, heat, and dehydration than about growth plates.
You should be able to figure this out just by how your pup responds.
We've never given the lepto vaccine to our dogs but vets in our area are currently recommending it because of some recent outbreaks. We gave both our adult girls a shot recently -- never had them before -- and I think that our oldest did develop some black scale on her ears because of the shot. No way to know of course, just a coincidence and our vet couldn't find a cause.
In any event you should check with your vet since the need for a lepto shot seems highly geographically variable.
I think @Zande has more or less covered things. Not sure about the clingy part. We had a female who, the first time she went into heat, carried on and on and on. So much so that we started singing the Annie Lennox song "I Need A Man".
Generally a female will be in the estrus period for 1-21 days (or longer). This is when she'll have red blood cells in the reproductive path. She will be most fertile 4-6 days after the surge in the luteinizing hormone signals the eggs to drop, which might be 7-10 days after you see the first signs. Without testing there isn't any way to know when the luteinizing surge occurs.
I also think the some protection for the furniture would be a good idea. We didn't find disposal diapers useful for several reasons but have used Simple Solution Washable diapers with Wee-Wee Disposable Diaper Liners to good effect. Have never had a problem with staining on the furniture. She will most definitely want to clean herself so you'll want to give her an opportunity to do that when she's somewhere that minor issues won't be a big deal -- before or after walks, when she's in a rooms on her blanket, in her crate, etc.
Finally, male dogs seem to pay more attention to pee areas than to the actual area of the female dog responsible for the peeing. Consequently if you are worried about male dogs take your girl out for her to pee -- and she'll need to pee more. Actually I'd recommend this even if you aren't worried about male dogs.
Overall it's not a big deal so don't stress too much about it. It's easier than a bad case of diarrhea for sure.
I think this dog needs a home without children. While Basenjis aren't necessarily great with all children, they are usually great with children from "their" family. For whatever reason this isn't the case. Consequently this is definitely not a good situation for either the dog or your son. To protect both of them it would be best to find another home.