"In order to get along with a Basenji, you have to be at least half as smart as the dog!"
I wouldn't wake a pup to take it out, but what I did do is sleep right next to my second pup (in her crate) the first few nights. I am a light sleeper, so was aware when she would wake and took her out immediately. After those first few nights she slept with me, so I was very aware when she woke. I got her at 7 weeks (yes, early, I know) and in a couple of weeks she was house trained. I put it down to religiously getting her out, so she did not form any habit of going in the house. My first pup was 12 weeks and I crated her. Very few incidents with her, either, but I didn't start her sleeping with me until she was quite a bit older. (I learned, and with the second pup I made that step very quickly because it works so well!) My others were rescues, so came already housebroken, but both slept with me from the first night. Basenjis love to be with "the pack", even if their pack mates have two legs instead of four!
How tall is she and what does she weigh? A side view standing would be useful. Is she fairly square like a Basenji? When she barks, is it one or two separated barks, or a series? Some Basenjis do bark, but usually just once or twice when startled, not a string of barks. (Of course, whatever the mix is could affect this). As stated, you need DNA to be sure. Maybe Lab, but it doesn't jump out at me. I used to know a German Shepherd cross that looked a bit like her, including the colour.
In my experience boots may be more needed than a coat, although my boy used to wear both in really cold weather. Without boots he was done very quickly, trying to hold up multiple cold feet. My bitches, OTOH, did very well without anything unless it was extremely cold, although boots were necessary when it was 20 below! In a pinch, socks cut to fit secured with elastic got them out long enough to pee!
The breeder I got one of my girls from had a similar situation, but in her case it was two bitches who had always gotten along well, were crated together at shows, etc. All was well until the younger bitch had puppies. The older girl had a very mothering instinct and unfortunately got in with the pups, which the younger bitch resented bitterly. There was a fight and the two bitches were mortal enemies after that. (the same older bitch had mothered other younger bitches litters without incident, with bitches that were not so possessive of their pups). My friend had to keep them separate after that, and she ran two groups of Basenjis for years, one upstairs, the other downstairs, changing the order up on a regular basis. Not the easiest way to run a household, but apparently unavoidable if you want to keep both girls. My breeder also said it was more difficult with bitches that didn't get along than with males.....
Although she definately looks BASENJI to me - the LIKEING OF WATER surprises me. My purebred BASENJI does everything to avoid water.
Out of the five Basenjis I have owned, two were fine with water, one mildly disliked it, and two really did not like it at all! My second girl would jump off the dock to pursue a water skier, so her chase instincts overcame any aversion to getting wet.
Growling may be a warning to leave the dog alone or it will escalate to a bite, or it may just be complaining. My first Basenji would snarl and scream if I picked her up when she was comfortable, but I just laughed at her and she never followed through on the threat. My second girl would "swear" if told to do something she didn't want to do, and would snark if you moved her when she was comfortable, but again there was never escalation. One out of the five Basenjis I have owned would follow up the growl and bite if he was upset, but generally I have found that it is usually just their way of registering displeasure. Resource guarding can be tricky to deal with, however, and may result in biting if the dog is serious about retaining "his" possession. This gets dangerous if he gets hold of something that could be harmful, so personally I don't like to let it go. My own feeling is that if the dog has something that is indeed "his" that has been given to him, e.g. his dinner, then I won't provoke any confrontation, but if he has stolen something then I will take it from him immediately, no playing games, no "trading". The last thing I want my dog to learn is that stealing items is a good way to get rewarded with treats!
Regarding personal space, I have generally tried to finesse this rather than confront it. Many Basenjis will snark if disturbed when they are comfortable, and in my experience this does not change with time. If I need the dog to move, I may issue a command, but what has generally worked for me is to sit down next to the dog and gradually, with honeyed words and patting, maneuver said dog out of the comfy spot. I can usually avoid direct confrontation and manage with barely a grumble.
I'm not clear on the "problem". Is she defecating in the house? Usually you can tell when a house trained dog wants to go out, as even if they haven't been taught a clear signal they get restless, perhaps run to the door or whine. If she isn't house trained (at 4 years old??), you need to be observant and get her out if she shows signs of needing to go, and crate her when you can't observe her. Take her out immediately when you release her from the crate. She should get the idea, but with a Basenji you may have to accompany her outside, particularly in inclement weather. They can be stubborn when it's raining! In difficult cases, i.e. she refuses to defecate, using a match may solve the problem.