In cities, most everybody who’s wearing shorts, flip flops or white sneakers is a tourist, but since most tourist destinations are swarming with tourists you certainly won’t stick out.  Europeans tend to wear shorts when hiking or  “outdoors” but not in cities.  You may find some churches that want more leg covering, although if your burmudas are almost knee length you probably won’t have a problem. 

You can run into very cool, rainy weather almost anywhere in Europe, even in the mid-summer.  I was in Paris one July day when I was thankful for long sleeves and a jacket and would have been very uncomfortable in shorts.  Conversely, when there’s one of the used-to-be-rare-but-increasingly-less-so heat waves it can be sweltering, especially since air-conditioning isn’t all that common in Northern and Central Europe.

bermuda shorts are the type of garment in which you will always look like a tourist.

there’s nothing wrong with looking like a tourist
just be comfortable
if you want to go into St Peter’s in Rome you gotta have pants though
my travelling clothes for bottoms were 2 pairs of shorts, 1 pair of khakis, and one pair of zip off pants.

If you’re set on the Bermuda shorts and Pumas I think you’ll be fine.  You’ll find that most young people in Europe are fairly casual, so assuming you’re young you shouldn’t stand out.  Pumas are pretty popular in Europe so you should be OK.  I’m glad that you’ll be wearing fitted tee-shirts, these are key in preventing you from looking like a dowdy American tourist. 
So rest assured you can be comfortable and still look good.

Perhaps consider bringing a dressy top/tunic to wear to smarten up your Bermuda, flip-flop ensemble.  Also wedge heels are really popular now and they can be fairly comfortable for walking, these will also dress up your shorts and tee shirt.  Check out Travelista if you need any more packing advice!

So what about pumas and shorts for guys? Shorts and dark shoes look pretty goofy if you ask me, but what is the alternative if its really hot outside and you are on your feet all day. I’ve been told that white trainers/sneakers will definitely stand out, but won’t khaki shorts and black shoes stand out as well? I’m really struggling with what to pack here (trying to get the best combination of comfort, versatility, and inconspicuousness: jeans or khakis or dark slacks, plus dark pumas or some kind of trainer…
I am for certain bringing dark, fitted t-shirts and two (maybe one) pair of khaki shorts, and will be in Spain, France, and Italy May 23rd-July 10th. So keep that in mind when considering footwear, because I’ll most likely be wearing that in those places/temperatures with the shoes.
Thanks for any help you can give.

If you want to wear shorts and white sneakers, wear shorts and white sneakers.  You’ll “blend in” just fine —allbeit with the other tourists.  If you are of the (probably mistaken) opinion that you can actually look like you’re local, then don’t wear shorts and white sneakers.  Most of the places tourists visit are cities and most Europeans don’t wear shorts and white sneakers in cities.