The only antibiotic that has ever been shown to interfere with birth control levels and effectiveness is a medicine called rifampin which is used to treat tuberculosis. Rifampin may also interfere with the birth control patch and vaginal ring so if you are taking it, be sure to use a back-up, non-hormonal (i.e. condom) form of birth control.
There are some other medications that can interfere with your birth control, however, and if you are taking any of them you should always use back-up contraception:
Griseofulvin
Phenytoin (Dilantin)
Carbamazepine (Tegretol)
Phenobarbital
Topirimate (Topamax)
St. John's Wort (herbal supplement)
But in general, your birth control will not be affected by any run-of-the-mill antibiotic that you might be taking for things like sinus infections, strep throat, urinary tract infections, skin infections, acne, etc. Some people believe that because antibiotics disrupt the normal bacteria in the gastrointestinal tract, they will interfere with the absorption of the birth control pills from your stomach. But this is not an issue, and even if it were, it wouldn't apply to birth control methods that don't involve swallowing pills like the skin patch or the vaginal ring or Implanon.
Please discuss the question about birth control with a pharmacist or your doctor, I do think some of the antibiotics are said to interact with birth control pills in a way that makes them less effective, but you need to hear this from a doctor or trained pharmacist, not the Internet.
As far as stopping taking the doxycycline, ask your doctor about that.