We are staying in Begur in June, and want to make a day trip from Begur to see Dali Theater in Figueres, then stop by the Monastery St. Pere Rodes, then drive down to Cadaques, walk around, have lunch (perhaps at Compartir), see Dali Summer home and drive back home to Begur.
I would appreciate the input if this is a reasonable trip or is it too much to try to fit in one day?

It's an easy day of driving, covering about 100 km one-way, but the roads will be busy. Make sure you have reservations to visit the Dali Theatre in Figueres, and don't forget about visiting Roses and the Roman-Greco ruins near L'Escala (Seu Empúries - Museu d'Arqueologia de Catalunya - Seu Empúries) before returning to Begur.

Compartir is open for lunch in June, except on Sundays. Reservations highly recommended (COMPARTIR).

How is the drive from the Monastery to Cadaques? Not too difficult, I hope?

You also need reservations to visit Dali's home in Port Lligat. Cap de Creus is a beautiful place to visit, too, but I don't know if you have time to include it in a day trip.

Thank you, have made reservations at Compartir and Dalu House to follow.

I wouldn't describe the drive as particularly difficult, but I encountered some incredibly fierce winds around Cadaques, winds that made it very hard to control my very small rental car.

I give details about this day trip we did in 2017 in my BCN/CB TR: LONG Trip Report Barcelona & Costa Brava June 2017
scroll down to the CB portion and you'll see this day trip. However, didn't make it to the monastery!

Regarding the roads around the Monastery and Cadaques, they are steep, but wide enough for two cars to pass each other without effort, so nothing too difficult. Have a look on Google Streetview and judge for yourself!