The development of a stye (or hordeolum) is a very common problem and often is not associated with any negative consequences. The main concern is if the stye develops a secondary bacterial infection, which would necessitate a more urgent medical evaluation. I would start by applying warm compresses to your closed eyelid for approximately 10 to 15 minutes. If the stye does not improve within 48 hours or if you develop redness and swelling that extends beyond your eyelid to your cheek or other part of your face, I would visit the emergency department if you cannot see your doctor
within the next day. Otherwise, I think you are safe to wait until your appointment next week.
In terms of your eyelid twitching, called myokymia, this again is very common. It usually affects the lower eyelid and can be the result of a number of causes, including stress, fatigue, dry eyes, allergies, caffeine and eye strain. The best way to treat this condition is by identifying the underlying cause and rectifying it. If you cannot identify any of the above causes as the culprit, I recommend getting plenty of sleep, avoiding caffeine and alcohol, and appropriately alleviating any stressors in your life.
Neither of these symptoms necessitates you going to the emergency department, and I think you are safe to wait until your doctor's appointment next week (except if you experience the concerning symptoms detailed above). However, if there are other concerning symptoms you are experiencing that complicate this matter, it is always better to be safe and go to the ER than to ignore the symptoms.