How many days should I spend in Jeju and Seoul?

Yes, three days in Jeju sounds like a great sweet spot for a 9 day trip. I would suggest you pick one half of the island(the west half is better for first timers I would argue), since it won’t be enough to cover the entire island.
 
If you are coming in May, it’ll be pretty warm in Jeju – you could go swim in the beaches potentially, or enjoy scuba diving, surfing, eat lots of sea food, and hike Hallasan, or many of the large hills formed by volcanic activity. Jeju is a very chill, outdoorsy island so spend lots of time outdoors Here’s a good guide I found on activities.