My experience in your planned countries, traveling on my own, is that English is fine. Polite, with a smile and good manners English. I had been worried about English in France, since at the time I was there the US and France politicians were in the midst of one of their periodic rather nasty spats. But found I was always very kindly and helpfully treated. Far beyond mere politeness and correctness.
Usually, for trips like you are contemplating, I start with some good guide books and my computer and make a list of all the things I really want to do. When short on time as you are, I resort to "top attractions in x-city" and see what I simply can't resist. Then plot the activities on a map. The bases tend to suggest themselves. Although it has been many years ago, I once found Avignon a great base for my choices. Sometimes right in a city is plenty to fill several days.
My usual start on hotels is
www.booking.com and I'm careful with the reviews. I do like the AirBNBs as to me they add another level of authenticity to the trip. But I read ALL the reviews. For hostels, I stick with Hostelling International. I'm sure there are good local ones, but I don't have the resources for checking them out. Hostels are also my last choice at my age, only when hotels are really expensive and AirBNBs scarce. An advantage of the hostels is meeting other travelers with similar interests.
Also like to shop in local markets and cook a bit. Different ingredients and flavors. AirBNBs make this possible.
For train travel, there is wonderful information in
www.seat61.com. And it has links to further information on the country specific trains. For short hops, it is usually easy to just go to the train station and buy your ticket via either a machine that almost always has English, or a window where the person usually has English. I find the machines easier as I can move a little slower and check things carefully.
Buses are trickier to research, but there seems to be a fine network of them. The one hard lesson I learned about buses in Italy is that there can be either very few or none on Sundays and some holidays.
My starting point for foreign travel transportation is usually
www.rome2rio.com. If you aren't familiar with it, all you need to do is put in an origin and a destination and it gives you information on all the reasonable modes of transportation. For very short trips it includes walking and biking. For longer trips it includes air. It doesn't always have the absolute best possibilities, but always presents one with good possibilities for general planning.
I see Madrid to Perpigan is about a 4 1/2 train ride right in the middle of the day, and you get to enjoy the scenery.
I like your idea of picking a "base" and then exploring out from that in all directions. Walk, rent a bike, bus. Using rome2rio, I can see it is only a 20" train ride from Arles to Avignon. An easy day trip. On the other hand, Avignon to Aix shows as 2'35". But the train goes through Lyon right about in the middle of the run. Almost suggests Lyon as the base, and then day trips in both directions?
I haven't spent time in the Bordeaux area, although it is on my long list of possibilities. Hopefully somebody else on this site will chime in and offer some thoughts.
Your trip is sort of suggesting Madrid-Perpignan-those A cities-Bordeaux-Perpignan-Madrid. Unless you can head home from Lyon or Bordeaux? Or maybe only keep two of the three optional destinations (Perpignan, Lyon, Bordeaux)and do those more intensely?
I do try to always spend at least 3 nights per location. The travel from point to point takes up quite a bit of that day. So for a 3 night stay one only has two full true days at a location. Also, when I arrive, I at least try to buy my departure train ticket right away. That way, if I run into some issue I have time to resolve it. Strikes, track outages, holidays, ??
In my prior post I forgot to mention Perpignan's
Palace of the Kings of Majorca. I found it impressive and interesting.
I wandered a bit here, but I was enjoying just thinking about your trip. When I'm not traveling, my next favorite thing is thinking about traveling.