"In order to get along with a Basenji, you have to be at least half as smart as the dog!"
Basenjis usually come into season once a year, typically but not always in the fall. You are confusing human biology with canine, dogs don't have "periods" as women do, they have heat cycles (coming into season) which is the equivalent, and in Basenjis that will be once a year, when there will be discharge so you will be able to see when it happens. I am assuming from your question that your parents want to wait to spay her until after her first heat? That is a good plan, IMO.
Recall and off leash are debatable subjects. Of five I have owned, my first was solid on recall, my last also was good but I did use an e-collar to ensure it, which is frowned upon on this forum, my second girl was good out in the wild, not so much in familiar territory, and the other two were unreliable. I wouldn't count on reliability off leash. Depends on the individual dog, and even more so on your relationship with said dog! On the beach or woods can be doable.
With my first Basenji I treated her like any other dog, walking her off leash in the city of Toronto! She was trained (as were my other dogs) never to go on the road unless at heel. Seems incredible looking back, but at the time I didn't know you can't do that with a Basenji, and apparently neither did she!
Long walks are great if you have the time for them, but not really necessary if you can provide play time. Two definitely will do better than one if you aren't home for long periods of time. My girls were good company for each other, easier to leave them than when I had only one, although my last boy was fine on his own once I got him through the initial separation anxiety, the key being no crating!
In winter it gets pretty cold here, and long walks are not really practical when all your dog wants to do is go home! Mental exercise works well, and I spent time with Perry teaching him to listen and separate the words that pertained to him from rambling speech, something that was fun for both of us and kept him engaged. Hiding his toys or playing hide and seek were also fun. Yes, Basenjis are independent in their own way, but when they decide they want to be with you, they can be very persistent! Privacy can be hard to come by when your dog doesn't tolerate a closed bedroom door.
You might bear in mind that a mix may have characteristics or allergens that a purebred would not have, and also that a "mix", unless DNA tested, may not even have a drop of Basenji blood at all. If you have selected a Basenji as your ideal dog based on your research, perhaps you need to concentrate on finding a "real" one?
A couple of thoughts. Although many Basenjis do not like water, most can swim. It's a good idea to give your dog some (possibly unwanted) experience swimming so there won't be panic if they end up in the water. The life jacket is a good idea, especially if you will be any distance from shore, but getting an overboard dog back into a kayak could be tricky.....best to practice this at some point to find out how difficult it might be and how your dog reacts. If your dog doesn't like water and understands the risk of going overboard it may be more cooperative in the kayak. My Perry was good on my paddleboard after the experience of ending up in the drink, because he detested swimming and was careful that he didn't risk another dunking.
Has anything at all in your household changed coincidental with his acquiring the cough? Sometimes something supposedly innocuous can actually be the source of the problem, and it's often something you wouldn't even think about, e.g. you are using a different shampoo on your hair, a new body lotion, you got your carpet cleaned, whatever....
Make sure the water is fresh, if it's been sitting around all day you should change it. I concur with Zande about giving meat or canned food, not only dry, and definitely mix in some water, let it absorb for a minute or two before feeding. Personally I would ditch the dry food, but that's up to you. And I wouldn't leave food down. If she doesn't gobble it up, take it away and offer it later. Nothing like stale food to create a fussy eater!
I have no idea how it will work with lymphoma, so don't get your hopes up. The veterinarian I was working with at the time told me that it might shrink brain tumours without eliminating them and we worked on the basis of suppressing symptoms which in Lady's case seemed to help. As they say, "your mileage may vary". I just wanted to alert you to possible side effects....