Should I take amoxicillin and/or bactrim for infected hangnail or can I take them together....And what else would you suggest I did bite my nails very bad until now but there's no pus as I drained it myself with a sterile syringe... thanks for any advice

Hangnail. Hot soaks with Epsom salts would work better than using either amoxicillin are bactrim (sulfamethoxazole and trimethoprim)
Skin infections. Warm soaks and after patting dry,apply antibacterial ointment like neosporin. You should not need both oral antibiotics-one is sufficient. If pus accumulates or redness extends,have a physician evaluate and treat you as doing your own drainage procedures is not advisable. Also any purulent material should be cultured since you have been taking antibiotics which may not be effective.