First, welcome to the basenji forum! You'll find that, just like raising kids… everyone here approaches basenji training a little differently.
As you pointed out, your lifestyle is unstructured (so is mine :)), so if you are trying to apply a schedule to Dexter, and if it's not in your nature to follow a schedule... then you may end up being inconsistent in your approach.
In my opinion, consistency is more important than following a set schedule. Having a designated word/phrase to get him used to going potty on demand is important. The other thing important to basenjis is HIGH PRAISES when they've done the deed. Again, you want to reinforce the positive and ignore the negative. Negative reinforcement when accidents happen is the worst thing you can do to Dexter! When puppies potty in the house, it's not because they are being spiteful, it's because they are excited/overstimulated/or they just simply have to go.
(Question 1- How long do I give him before I bring him back inside if he isn't pottying?)
My advice: be patient…. if you've read his body language and you think he has to go potty, stay outside for as long as it takes. Keep repeating "go potty", and when he does, lots of high praises and excitement.
(QUESTION 2-I noticed that the place that we were bringing him was where our neighbor's dog used to potty before he got stolen- is that why he didn't want to potty there? because it has another doggie's smell?? idk)
Basenjis love to sniff, sniff, sniff. Is there a reason why you want to pick out his potty spot? I'd let him sniff and find his own spot.
(QUESTION 3-How long before he should stop whining in the crate? I know that we are supposed to be building his tolerance for it by putting him in for increasing intervals, but I don't know when I will see the effects of this.)
Crate training is fickle, and has to be done in the right way…. again, according to his style. You don't want the crate to be a prison, but a place he wants to go to. If you read some of the threads on here, you'll find some great advice on how to get Dexter used to the crate.
ALSO: We are bringing his crate to out bedside at night for bedtime. He is sleeping in there right now, my fiance is sleeping near him. He seemed to only whine for a few seconds before falling asleep. (QUESTION 5- Should Dexter go to bed when we do, as we rarely follow a schedule ourselves, so he may not want to be sleeping without us there? Also, do we continue moving his crate from bedroom to living-room and back each day for good?)
Some people don't want dogs sleeping in their beds, but basenjis love to cuddle. Many people on here sleep with their dogs, but I'm sure there are many on here who keep them kenneled at night. My thought is… if he is kenneled all day, let out when you get home, and then kenneled again at night... that might be a lot of kennel time for the little tyke. You'll have to decide.
I also wouldn't move the crate around. Just leave it in one place, with the door open, so he can visit when he wants to.
FINALLY: The whole point of getting Dex accustomed to his crate slowly and comfortably is so that he will not despise being in it when we are not able to be home with him and he will not potty in his crate (as long as we let him out in time for a bathroom break), so that means that at some point during his "getting used to the crate" process, we will need to leave him in there for a while when we are home as part of the process, but if he is crying to get out- do I leave him in? Take him out because I read in the book that if he whines in the crate that we are moving too fast with him? And if he is crying not because he has to potty but because he wants out, but it IS potty time, do I take him out to potty and risk him thinking that it was the crying that freed him?
Lastly, basenjis can't read. Dexter doesn't know what how he is supposed to act! They are social creatures and often want to be where you are, so being locked in a crate for training purposes may feel like punishment to him. I don't use a crate, but there are lots of threads on here that may point you in the right direction. However, you want the crate to be his cave-home, not a punishment cell.
Again, there are no right and wrong answers… you, your fiance, and Dexter will learn together! Remember to be consistent, with lots of positive reinforcement... and to enjoy! They grow into their wrinkles so quickly!:)