Sty removal question - does the eye doctor have to cut into it?

Yes and No! Depends on many things! Size, location, how "hot". .. If a pus head is presenting, a simple incision is easiest. If it has become long standing and firm, it may need to drained or if small, treated by a steroid injection! See an ophthalmologist.
Stye treatment. Usually sties resolve on their own with simple application of warm compresses (for 10-15 minutes 2-4 times a day) to bring stye to head & rupture (do not squeeze it). Recurrent sties may require antibiotics. If stye doesn't drain completely, then the doc may have to lance it (after using local anesthetic) to remove its contents & speed healing.