Should I work out at the same gym that my ex works out at?

Absolutely. Outlast them at the gym. Let them see the body they can no longer "get". Let them see the transformation. My ex-wife looks gorgeous, but then again, so do I. We will not allow our bodies to get out of shape, even though she posts her photos on FACEBOOK and I rarely do. I am not focused on showing them what they are missing, rather, I am showing how much I am enjoying my life and taking care of my body and mind.

Do not let your mistakes from your past deny your destiny for a better life.

Is it more important to get 8 hours of sleep but sleeping late or sleep earlier?

According to a research , conducted in japan sleeping more then 8 or less then that induced high risk of diabetes , blood pressure and a higher body mass index. And those who slept between 6.5 and 7.4 hours a night had the lowest A1C levels of all the participants. A1C levels are measurement of a

Is it safe to exercise 20 minutes after waking up in the morning, before breakfast?

Yes, fasted cardio is perfectly fine and you won't run out of your glycogen source if you only exercise for 20 min (even if it is 20 min HIIT). Your post workout meal (breakfast) will also quickly replenish your glycogen source if you are low.

How to manage to get fit, healthy, skinny as a teen

You're young, so don't worry too much about it as you're still growing. A simple solution is to just take up a sport. It doesn't even have to be a team sport, just something that compels you to be more active on a weekly basis.If what you really want to do is add on muscle and build your physique: