I may not know as much about APs as some of you, but I do know a fair bit about fitness and nutrition so I'm going to add a few notes here:
Everything you said is great, however I would suggest to cut out sugars altogether if you're seriously wanting to lose weight. Sugar is the main culprit for weight gain, or lack of weight loss. So this means no soda at all, not even the "sugar free" ones. The fake sugars they have in them still trick your body in to thinking it's had some sugar. Drink water or herbal teas.
Also, lifting weights definitely burns a good amount of calories, as the body keeps burning calories even after you finish working out, as the muscle fibres are repairing. However yes cardio also burns calories, obviously. If you want to get serious about burning calories, do a weights workout then some cardio immediately afterwards. But I am acutely aware that a lot of people in this thread won't have the energy or motivation to do that due to anhedonia.
If someone wants to pick the best form of exercise for weight loss whilst also experiencing anhedonia I would suggest interval training, doing sprints, running up sets of stairs, doing short bursts of high intensity activity (High Intensity Interval Training, aka HIIT). It makes the workout feel like it's over faster, and it gets your heart rate up and gets you sweating real nice, which = calories burning.
This may be true but it is also pretty expensive and not a viable option for a lot of people, especially if they're unable to work due to their side effects.
I have to strongly advise against tanning booths and sunbaking. Both of these can cause skin cancers. Yes, daily sunshine is really important to boost your mood, and it is vital for vitamin D production in the body, but you only need 10-15 minutes a day in the sunlight for these effects, not extended periods of time.
I would advise against using the BMI to measure your appropriate weight. It is an incredibly outdated and inaccurate scale and there are a LOT of body types that it simply does not measure accurately. It consistently causes very skewed results and should just be avoided.
A better way to measure YOUR PERSONAL WEIGHT LOSS is to record your body measurements once a week i.e. waist, chest, thighs diameter, take before and after photos and compare them, consult with your doctor for advice. If your doctor refers to the BMI they are just being lazy and/or uneducated, so see a different doctor.
Plus alcohol is just pure poison, it does SO much damage to every single organ in your body, bar none. And it is VERY high in calories!! If someone is trying to lose weight, but is drinking alcohol, they will not lose weight. Alcohol contains a surprisingly high amount of calories. Keep that in mind everyone!
I'm so sorry to hear about the stress your own mum is putting you through. Why is she doing that?