For years I wore boots, it was just the accepted thing, then I heard about approach/trail shoes. I was suspicious for a while, but I was persuaded to try shoes by two incidents fairly close together - I talked to a couple of people who had done LEJOG in shoes, and I read Ray Jardine's books
When i did get a pair of trail shoes, I was pleasantly surprised and impressed
I still wear boots for high rocky hills, like the Lake District, and in winter, but for all my walking now on lower level paths and trails i wear shoes. Indeed, i have all the enthusiasm of a convert, and sometimes accost strangers whom I see wearing boots, and preach the gospel
Last summer I wore shoes to repeat a section of the PW that I had done years before in boots, and had no problems at all
However, I think that something like Innov8 or fell running shoes might be just a bit too light, both for support & protection, and they will, of course be liable to wear
Something slightly more substantial , but still a shoe would probably fit the bill. I have settled on Merrell MOABs; the fit , comfort, and support is just right for me
I suggest going to a shop and trying a few on; many shops will have offers
I wouldn't bother with waterproof socks or membrane shoes. the shoes are so light & comfortable that there's no problem if they get wet, and they dry quickly
If it does look as though the rain is set in for the day I just use a plastic bag over my socks - the ones that come with sliced bread are about right