Kathy, I have rather high arches, too. Before my long 2015 trip I went to an expensive shoe store (not my usual source), and the salesman recommended Aetrex memory-foam orthotics inside the lace-up shoes I had selected. I used the orthotics to replace the much thinner inner soles originally supplied with the shoes. The orthotics provided some additional cushioning as well as arch support. Unfortunately, they are expensive. There might be less expensive options that would also be helpful.
There's no substitute for trying on a bunch of sturdy shoes to see how they feel. I walk a great deal and take only a single pair of shoes, so I love having something with a very spongy, cushy sole as shown in that link, but I don't think any of those shoes would be great for rainy weather. They are very lightweight, however, being made partly of (very porous) fabric. The open-work loops on the soles provide the bounce I love, but they are far from ideal for country walking.
It seems that where shoes are concerned, there are usually trade-offs--and that's before you start thinking about the cost of quality footwear.
I took leather-soled sandals on my first trip to Europe and learned from the experience. I would not want any kind of thin-soled shoe in Europe because your feet will be on hard surfaces almost all day long. Perhaps simple flats would be OK if you plan to change for dinner and will be eating in your hotel or taking a taxi to and from the restaurant. Otherwise, I think you'll find that the bottoms of your feet hurt.
