2 months overseas with toddler

I might be out of my mind but...

DH will be deploying this summer and I just found out that our landlord will not be renewing our lease when it is up (I guess the house is now in foreclosure). Im now thinking about putting all of our things in storage for a few months and going to live overseas for a bit with DD while DH is gone (she will be 27mo). 

Where would you go? I'm leaning towards somewhere in Europe at this point. I am thinking that it would be better to base myself in one place and do day trips from there (maybe stay in one place for 3-4wks and then move on to another country?). We have seen a decent amount of Europe (Paris, Rome, Ireland, Turkey, London, Czech Republic, Munich & Bavaria) and I am not against going back to those areas....or I could branch out and try something new. DD has been to several of those places with us and travels great.

I am hoping to find a 1-2 bed apartment for no more than 500E/wk 

Don't want to have to worry about having a car....I do not plan on bringing DD's carseat and renting one makes me nervous. Since I will be by myself I only want to take a suitcase with both our clothes/toiletries, backpack, Ergo carrier, and our small umbrella stroller!

Want to be based in a nice place that we can relax and enjoy the local culture while having good transport options for several day trips.  

I want to be close to good medical care since DD will be with me and there is a chance I could be pg during the trip (we plan to TTC right before DH leaves....). 

Any suggestions? I would love to spend more time in Italy but Tuscany needs a car and Venice doesn't sound like fun with a toddler   I'm thinking of doing Sept-Oct since the main summer crazy will be over!