I am also a a foreign (South Korean) entrepreneur and was looking for investors in both South Korea and the US. Most of our members are college students in the US and we are planning to deploy a service starting in Korea and then US West. My suggestions are based on my personal experience.
Question 1. At what stage do I have to meet them?
It depends but I believe frequent contacts do not harm you as long as most of prototypes and concepts are ready. We were not ready in prototypes when we started to talk with investors, but at least we had concrete concepts and fairly great profit model. As a result, we are about half-way to getting investments.
Question 2. do you think it's worth it to spend $2000 in travel expenses and yet it's not guaranteed to get funding?
I think that's risky to spend a big money on that.
Some investors are generous enough to have video conference, so you could suggest that.
All the members in my startup are scattered out in South Korea, US Central, and US East.
So it was hard for us to meet at the same time in the same place.
As a result, we have been using emails with the investors and will have video conferences.
So far, they say it is highly likely for us to get funded.
As long as your startup idea is great, I don't think it's crucial you need to spend travel expenses.
Question 3. What if i even had to repeat spending that for multiple investors?
Since money can be an issue, we kept looking for investors who we could meet through video conferences. I would physically visit to them if the progress is almost finalized.
Question 4. If i had a partner who lives in the US, Is he enough to get funding for us alone?
I am not exactly sure what you mean by 'who lives in the US'. But I saw a few cases who live in the US as a foreigner (not a green card holder) and get funded by US investors.