Gum swelling is normal with braces. Keep your teeth very clean. It usually goes back to normal after removing braces. Regarding gums popping out between upper teeth, you might require a small surgery called frenectomy. Your dentist can help you better. Do ask them your doubts

There may be deposits accumulation that lead to hum swelling ... might need cleaning o teeth along with mouthwash and regular brushing ..you can use water flosser

You need to invest more time in maintaining your oral hygiene since you have braces now. Bleeding and swollen gums are most likely due to plaque accumulation. Follow all the instructions given to you by your orthodontist. Use orthodontic toothbrushes and mouthwash.

As you  are undergoing orthodontic treatment need not worry about the gap between the teeth , that gets rectified by the treatment . you need to undergo regular cleaning by the dentist, as it removed removes the plaque and bleeding of the gums can also be controlled.

If you are already undergoing orthodontic treatment, the best approach will be to consult your orthodontist for these queries. Rational opinion is not possible without a look at the clinical condition of the teeth. Going by the description that you have mentioned, your gums will require cleaning so that the inflammation can subside, and meticulous maintenance by proper tooth brushing is equally important after that. Regards.