With only five days at your disposal and a pretty full wish list of places to see, staying in a remote location would appear to add unnecessary mileage to your stay.
If you were not averse to staying in a town, Figueres would offer good road links to all the destinations you list while offering the interesting Dali Theatre Museum on your doorstep.
Peratellada and Pals would be worth including on your itinerary to Begur.
Driving to Cadaques and back would require a full day due to the nature of the road.

I'd choose Girona over Figueres ifI had to choose one base for this area or if you want a coastal base Begur.

We spent a month in the area - rented two wonderful farmhouses for a week each, one in Orfes, near Banyoles and one in Anguillana, a tiny village on the French border near Figueras. We also spent a week in Cadaques and a week in Barcelona.

Banyoles itself was not very charming, but we loved the people there and it had all the services and supplies we needed. The surrounding area was stunning and it was well situated for all the day trips you mentioned.

If the property you found is truly great and you don't mind driving and soaking in the scenery, I would say to go for it.

Must-do day trips are Girona, Figueras, Dali's castle and Besalu. Cadaques, as others have mentioned is a little too far for a comfortable day trip (and it is fun to be there at night).

I just saw on another thread that you're going with kids - Our boys were 7 and 10 when we went, and they loved staying in a remote farmhouse w/pool, mountain bikes, etc.

Tossa de Mar would be my choice every time. You're not far from any of the attractions in the area and the old town is stunning.

Thanks for all the great suggestions.
We are strongly considering a villa in Begur now as a Spanish friend seemed adamant that we should be closer to the beach.
She felt that we would spend too much time driving if we stayed in Banyoles.