International Travel – Obtaining a European visa as a foreigner in the United States

First, let me explain my background. I arrived in the United States with a tourist visa 18 years ago from South Korea and have never been out since. So, my US visa has expired a long time ago, but I changed my status to F-1 (note the subtle difference between the F-1 status holder and the F-1 visa holder.) had no F-1 visa). However, I recently received a green card.

I am now in my fifth year of PhD and will be applying for postdoctoral academic positions.

Here are some questions:

1) If I travel outside the United States, back to the United States would not be any problem since I got a green card? I will not go to Korea because I did not do my compulsory military service; therefore, they will probably never let me go (especially since I left the country on a tourist visa and have never returned). However, I would like to know if I can travel to places like Europe.

2) I would like to have options to apply for certain jobs in Europe. However, for the reason that I explained above, I can not get a visa directly from South Korea. Is it possible for me to obtain a European visa (as a Korean and American permanent resident) by any other means? Maybe I can enter a European country and change my status to a valid legal status without obtaining a visa? (Just as I did in the United States with F-1 status)

EDIT – I have a valid South Korean passport, which expires in about 8 years.