I'm starting a new business and I found a foreign company with a trademark identical to my intended company name. Options?

It is impossible to give a definite answer to your questions because the USPTO has the final say on its approval process, but in my experience, I have been successful in obtaining a domestic U.S. trademark when there was a similar mark outside of the U.S. It takes a well thought out application to give yourself the best chance.

As for registration in the EU (or outside the U.S.), you would need to determine where this other company actually has a presence and avoid those countries.

It all depends on how you draft the application because there are classes and descriptions to carefully consider. I strongly suggest that you hire a Trademark Attorney to apply for registration. Let me know if you need help.

Q1 - Is the USPTO going to veto my application based on the existence of the other company's domain name and foreign trademark?
ANSWER: If neither the other company nor anyone else is using in the US the same or a similar mark as to the mark that you want to use then -- No.

Q2 —Assuming that the other company has a trademark outside the US but not in the US — can I apply for a trademark in the US without fear of the other company's future rights to the same name here in the US?
ANSWER: Yes. (There is only one exception I know of and that case was decided in favor of the foreign mark owner because the mark was famous and used quite near the US border).