I have read that basenjis are great in apartment situations because they choose to hold their pee longer than others. In my case that is very true. I don't live in an apartment and have had yards but both of mine would rather make me walk them instead of going in the yard. It was not until we bought a house with a half acre that they would even consider going in the yard and I had to make/train them to do so. I guess they don't like to dirty their living space fine with me.
Mine have only gone in the house in severe emergency and I felt my female I raised from a baby was very easy to train. I did not train our male (my boyfriend did) but he is just as good at never going in the house, even being alone for 7-8 hours. The breeder had already had our female box trained before I got her at 8 weeks so it was an easy transition. Start with the box always being near them when they play and sleep, You need to watch them and see the signs, you need to place them in the box sometimes.Then after they gain more control of their duties, move it to the door, when they go by the door and have not started to pee/poo yet, take them out. Then after they are used to that, remove the box and watch carefully for when they go by the door. Mine started to scratch the door, the male however used to stand at our feet or stare at us... He will now also go to the door. The one emergency was I away from home for most the day with a friend to walk her in the middle of the day and she got diarrhea. When I got home I could see she chewed at all the door handles, ripped down some window blinds, and tore all the jackets off the door hooks trying to get out. I felt so bad for her, she tried so hard I have not heard of other breeds with this behavior.
Rain and wet ground is an issue with basenjis, I will agree to that 100% - I had to force them onto the grass and tell them to go pee, go pee, go pee, go pee; in their more mature days, its not as hard. They will hold it in for hours and hours and stay in bed all day if you let them when its raining!
My two cents about biting, my boyfriend's male can get really rough and I don't like to excite him too much in fear of feeling the pain. My boyfriend and his two kids had always roughed house with him like that so I blame them for raising him that way... I always warn kids, don't mess with him and DONT RUN AWAY, he will get you. My female is very mellow and I have not had any issues like that with her, I trained her that way I would play with her and let her use her mouth but I would TALK to her and let her know when it was too hard.
And YES, never chase after them. I call their names so they look at me and run the opposite direction or pretend to bend down and be interested in something on the ground, works like a charm. You get them to play YOUR game.
Positive reinforcement and deterring bad behavior is the only way, Worked like a charm for me. Say, "no, that is my shoe, here play with your toy or chew your bone." Hand them the bone/toy.
My biggest warning for hopeful basenji owners is consider they are hunters and like to chase. typically, you cannot have them run free in any situation. They are not car/road smart and might run too far away chasing an animal. I used to have a nice, safe beach I would let my female run on 3-4 times a week and I trained her to come on command with the use of homemade chicken jerky, but typically, I think most basenjis will only come back to you when they are ready based on what I have read. I think I had spent so much time training my girl, she was sort of an exception . They are not like labs where you can have them free and they will safely follow you around.