Cool! Keep working on the false grip, you'll be muscle upping in no time! Are you doing horizontal dips?
Depends on what your fitness levels are. If you've never worked out for years/have practically no strength then obviously you'll have to start light. It's the same if you're working with weights. If you can't and you're really out of shape don't worry cause not every one starts at the same point. There are harder variations of bodyweight exercises and there are easier variations, you just need to figure out where your starting point. Do an honest assessment of yourself, ask yourself question like these: can you run (how many long, how fast...), how strong you are (can you do basic bodyweight exercises like pushups & squats). This will help cause like i said if your weak no point asking you to do hard exercises, right?

As for cardio vs HIITs, it depends on what you want out of them. While they both will help you burn calories (of course DIET is still the key for losing weight) they are both in an opposite spectrum; cardio boost your fitness levels and endurance while HIITs will boost your speed and strength. Basic bodyweight training will give you more strength. If general fitness is part of your goal why not do all.

If it doesn't hurt, ignore it. They say calluses are manly! LOL
Just remember to start slow and build up your strength especially when doing HIITs. Them HIITs can take a toll on your joints, tendons/ligaments if you're not ready for them especially on higher levels. Don't forget some rest time too.

BTW how does weight training disrupt your weight loss progress?    It shouldn't...