Your hymen doesn't "pop".
The hymen is a thin membrance of partially perforated skin across the vagina.
Most sports, running, jumping, horseback riding, gymnastics, etc can stretch the hymen as can using tampons.

Occassionally the hymen will "tear" this is the age old "bride bleeding on wedding night" rubbish.

The hymen most typically stretches - and some girls never have one in the first place.

Your urethra (where you pee from) and your vagina (where you have sex and bleed from) are not connected (well not unless there have been some severe and unusual circumstances).

Burning when you pee is most commonly a sign of a uti (urinary tract infection). These can be bought on by tight clothing, not drinking enough, holding on to pee when you need to go, amongst other things. Some people just get them for seemingly no reason.

Try drinking plenty of water (and cranberry juice is supposed to help) but if it hasn't stopped burning after a day or so you need to tell your mom so you can go to a doctor as you may need anti-biotics to help clear any infection up.

The bleeding may be as simple as some perfectly normal spotting.
From when you start your periods until post puberty it is common to experience spotting or slight bleeding between periods and nothing to worry about. Indeed many many women continue to get spotting between periods for much of their life.
It is possible if you have been doing the splits quite fast and or landing hard that you may have torn the hymen, even so it should stop bleeding within a day or so and should only be very light bleeding.

I think I answered (and filled in) most of your questions. If you have any more feel free to ask away and i'll do my best for you.