Europe is a big place, there are lots of areas with plenty of strays like Greece, and then places with no strays, like Norway and Denmark. Careful about making blanket statements about an entire continent or country and its people.
There are plenty of areas in this United States with no dog overpopulation that are massive sinks for stray dogs from elsewhere in the US. For example, New England, parts of Minnesota/Wisconsin, places in Oregon, Washington (and I'm sure others).
Of course, neutering your dog is your decision. Unless used for breeding or competition, I'd spay a female at about 2 years old. Breeding females, I'd spay after her last litter or around 7-8 years old. Males, it's more murky, but I wouldn't worry much about neutering a male at about 2, if that is your choice.
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