Need suggestions for a place to stay for up to 90 days so we can make small trips from there

Before we start, are you familiar with the Schengen Rules? You can stay no longer than 90 days in a 180 day period in the Schengen area. That includes arrival and departure days. All the places you mention are in Schengen.

It would certainly be doable to visit all of those places in 90 days, but not if you are trying to have a base in one location.

I think you need to answer some questions for better assistance.

1. Are you anticipating on renting a location for 3 months and also renting hotels throughout Europe for your travels?
2. Are you only looking at travels you can do in a day away from your base?
3. Would you consider several different bases over your 3 month period?

My first thought was Verona.

You are going to need a good transportation hub, air, train and bus. I would suggest that you focus on Paris. As previously stated, you need to take into account the Schengen Rules. Maybe limit yourselves to 85 days in the Schengen countries to account for any unforeseen circumstances; like plane cancellations etc.

You'll be paying double for your 90-day temporary residence and hotels in all those other places at the same time. So it's either point to point travel, staying one place a few days or staying put, living like a local and making a couple trips of 3-5 days plus train or air time. Or, you could travel around for 60 days and then stay put for 30.

Others have already commented on the obvious that it would be foolish to rent a place for three months and spend most of your time traveling and paying for lodging in all the widely listed places you mentioned.

How about splitting up your places for basing where you spend a week or two at each place?

For example:
To visit Tuscany and perhaps Umbria in Italy, you could stay in Siena or perhaps even Florence.
Take day trips to visit Orvieto, Perugia, Assisi, Lucca and more.

For The Naples Area, consider staying in Sorrento and visiting Pompeii, Capri, Naples and the Amalfi Coast.

For Switzerland, consider Interlakken to visit Lucerne, the Youngrfrau and other nearby places.

For the Loire, stay some in Porto and then move upriver to Regua or Pinhao for a few days.

For the Lisbon area, stay there and visit, Evora, Sintra, Obidos and more.

Visting the Loire Valley from Paris on a day trip would be a bit much, but base your self in Blois or Tours and visit the many Chateux.

Munich would be good for taking in Augsburg, Garmisch/Fussen, Berchtesgaden/Salzburg (however I recommend staying in Salzburg for 3 days).

Rome is worth a week and there are a few places you could do a day trip, like Ostia Antiqua.

For the coast of Croatia, the roads are tough so, visiting Dubrovnik and Split would be best done without bashing somewhere.

I would suggest staying a month in an agriturismo in Tuscany. Virtually every farm there has rooms and apartments for rent as an additional source of income.

Then I'd suggest staying outside of Budapest in a small town--alongside a train route.

Then fly over to inland Spain and stay in a place--like Granada. Apartments there are very well priced--as are restaurants and everything else.

Of course, you have to be out of the EU by the 90th day--including first and last days.