Are there any over-the-counter medications that are effective at treating cellulitis?

What's the cause? Cellulitis is basically a visible inflammation (itis) of the skin and soft tissues (red, hot, swollen and tender). It can be caused by many things including sunburn (radiation), trauma (sprained ankle) or poor vein circulation (stasis dermatitis). It need not be caused by bacteria. If you have repeated episodes of this, get to a wound care or vascular specialist, the diagnoses you have is wrong.
Possibly. Local cellulitis from a small abrasion may be treated with otc medications such as domboro soaks, topical antiseptics (betadine/peroxide) and different possible topical antibiotics. Care should be taken with neosporin as it can be reactive. All cellulitis should be taken seriously, although most out-patient clinics should be able to handle simple cases.
No. You must use antibiotic, but bromelain is useful to help resolve the swelling faster along with the antibiotics.