1) If you are moving from the US, you want to move to Europe not for better jobs, but for better quality of life overall. Europeans work less hours, have more vacation, and many more places to go to with a $50 plane ticket. Unless you live in NYC, your future European capital will likely be more diverse when it comes to culture and food.
2) Start by picking the city where you want to live. Scout it, see what is around, see what the people are like. Look for a job after you find that place.
3) Meeting people face to face will make your job search much easier. If you can AirBnB your US apartment, that might even be enough to pay for a trip to cheaper cities like Berlin or Barcelona.
4) Yes, you'll need a work visa, and while each country is a little different when it comes to those, generally being sponsored by a company is all you need.
5) In most cases, don't worry about the native language. I have found jobs, and know people that have done the same, in Zurich, Paris, Amsterdam, Berlin, and Barcelona, knowing only English. Once you settle, you might want to learn the local language for social reasons, of course, but there are plenty of English-only jobs.