Here in Israel, which is often considered one of the closest "startup capitals" to Silicon Valley, what almost any new company will do is just lunch in the US, since it's an easy market. Most wouldn't launch in Israel, since it's such a small market (8 million people).
A few will launch in Europe, but because of all the issues mentioned above (different laws, languages, cultures, etc), the US is a much more attractive first target.
Once they get big, they might then move to Europe.
That's by no means the only way, but it's a common one.
There are a few accelerators/etc who specialize in helping Israeli companies approach the US market and get exposed to Silicon Valley.
Other than that, most of the problems are kind of the same as with any startup - trying to build something people want. The biggest issue is if "things people want" is different in Israel and the States, and that's why exposure to the US market right from the start is so important.
As for immigrating to Israel - I don't really know much about the topic. If you're Jewish, it's incredibly easy, but if you're not, then I have no idea how the process looks like. Sorry.