Best Place to Spend 4 weeks Solo Working Remotely from Italy

I think it would be either Florence or Venice.
Venice may be a bit more beautiful to walk around but Florence has more great attractions to visit and Florence is more centrally located and has more nice towns to day trip to without a car. I would go with Florence.

Are you interested in a large city?
To visit Gaeta, I would think Rome (or Naples) would be your best bet unless some of the experts can recommend a smaller town that would give you a similar amount of transport access. But your budget might dictate the choice as well, as I assume it is going to be a lot more expensive in Rome. How about Salerno?
Otherwise, it is really a throw a dart situation!

I spent a month in Italy last winter: about 20 nights in Rome, 7 nights in Naples and 4 nights in Salerno. I enjoyed them all and didn't get through my (admittedly long) list of things to do in either Rome or Naples, despite intentionally skipping the Amalfi area and the islands.

Naples would probably offer much less expensive lodging than Rome, and there's a great deal to see in Campania. The annual Campania ArteCard would make sightseeing for the month very inexpensive.

If you prefer a smaller place, Salerno is worth serious consideration, but it doesn't have as large a number of local sights unless you want to make trips over to the Amalfi Coast.

Bologna would be a good location. Great access to small towns via local transit, excellent rail hub. Tons of things to do right in the city.

I vote Salerno too! (I wanted to refrain for a bit due to my zealousness haha) If you wanted a place to do some day trips from (Capri, Amalfi, Positano, Paestum, Vietri, even Naples and Caserta) that would feel more like real life and have lesser expenses, that would be a great place! Its a great place to stroll through the evenings along the water or even along the Corso and Via Mercanti with all the locals. I don't know of any particular cooking classes, but there are various markets throughout town if you needed (I love my flyflots I got at one that I use for house shoes, I call them the Italian old lady shoes), I like the "local" shopping (not catering to tourists, but stores that Italians from the region go to) and many great restaurants! You would start to recognize your local market folk as well as all the old men sitting around with each other playing scopa during your many walks!

Bologna seems to be a good idea too, its a university town, and apparently great food!

I admit I am a fan of northern Italy probably because I prefer mountains to beaches…. But Padua is an outstanding small city, near Venice and not too far from Verona, Garda, Milan, the Dolomites and Austria. 30 days only include 4 weekends for travel so sampling a range of options could be fun.

working during the week and being a tourist on the weekends.

Any issue with 'working' without some sort of work permit ( or visa ) ?

What a delightful proposition!! Usually it would be a toss up, but in late October / early November, I would choose the south. It should be rainy by then in the north. How delightful to have time to get to know people in the markets and around your home base!

You DO NOT need a work visa to work remotely, effectively you're working in the US, or where ever your company is located. I cannot count the number of times I've flown into Europe for work, even to France for 2 months, where I worked for a major bank, they would have gotten me a work visa if needed, it was not. I was visiting from an Asian office, no biggie, as thousands of people do every day across Europe.

Personally with your likes, I would choose a smaller base. I would pick Lucca. It is further north, but we were there in October a year ago and the weather was fabulous. From there you can day trip to Florence, to towns on the coast ( even the Cinque Terre) or down to some Tuscany towns.

I would also consider Siena, as to me it was an easier place to feel part of the local culture, but your daytrips will be more limited to Tuscan towns ( I think) and need to be by bus.

I just feel that with a city like Florence or Venice, when you are done working you are stepping out into crowds of tourists. While both the above are certainly touristy, they feel more like a place to live as well.