Yeah it is always the case the body being stiff when u just starting to learn proper breastroke. It is normal for ur leg to sink, especially if u are an athletic.
Ur bottom half sink is because of your leg muscle, that being said the more muscular you are you sink is higher.

You can further prove this by kicking ur leg away from the pool wall and without doing anything ur leg will sink but upper body remain afloat, and you ask another chubby person to do the same their whole body will float even without any kicking. Another good example is baby, their whole body will just float by itself.

For people who have sinking leg you have to counter it by kicking it from time to time, which is the hardest part in breastroke. Once you master it the rest should be easy.

You run out of breath is mostly because you panic while you are in the water, as time goes you will get use to it. Partly because you are not use to the water pressure when your body is surrounded by water.

1. This is more about getting used to and relaxing your body in the water. It is mostly about the techniques as time goes you will get the hang of it.

2. Yes you will have to kick your leg from time to time to keep you floating. What I normally do is is whenever I just wanted to stay afloat and on the water and not swimming, my leg would eventually sinks and whole body will be in a slanted position, to avoid this i will kick my legs so that my body returns to horizontal position, and keep repeating it. If you are swimming from one end to another you have to kick your leg anyway.

3. That have to do with timing yourself, if you timed yourself to have 0.5s to catch breath then make sure you suck in as much air as possible and exhale slowly in the water, when you ran out then repeat the same. One advice is do not hold your breath while you are in the water. The extra CO2 will drain your body and also have less time to breath in. The breathing out phase is always in the water and not outside. Let's say you allocate yourself 0.5sec to breath in, but if you are holding your breath and only breath in and out while your face is outside the water within 0.5 sec then is even harder for you to catch more air into your lungs.

As I know - yes. Actually wearing swimming cap is good, try it and become a good habits. 

For question 4:

- I do suggest swim after having your dinner about an hour later, do not swim with empty stomach, this will harm your body.
- Just eat some protein food (whole eggs, fish, chicken breast, boiled potatoes....) after your swim.

wish you happy swimming, 

1. Hygiene.
2. Less water drag on your head position.
3. Looks like more professional, 

I think one of the reason is that you may not engage your core muscles (abs).. many ppl thought your leg must constantly kicking so that it won't sink, but that's half.
Engage your abs all the time to straighten your body posture, and keep your hips high.
We don't just kick with legs, it often starts from our hips... and butterfly it may start all the way from chest (f not mistaken), and hips definitely.
So don't forget to engage your abs and hips!
My thought only la, i'm not a coach, and I have been swimming only for 2 months, but able to do all the  strokes. Somemore I may have mild asthma or near to asthma, but I don't have asthma!

For core muscles engage, there is one trick can do - extend your armpit move forward, not extend your hand move forward. Try it out, 

Do more stretching before and after swim.
its normal for your body to be stiff because u said that u do gym.

Swimmer have flexible body. have you see any pro swimmer are as fit as those that workout at gym ?