You can use "compete" for those instances :) "They are competing for the gold medal."

Yes you can say that. It is natural. We use "fight" for all sorts of situations that are not really fights at all. For example, "the sick person is fighting to get well" "she fought for her beliefs and stood her ground" *** "our team will fight to win" *** "he fought back tears"

You could definitely use fight. Some other words you could use are battling or competing as well. “The Korean and Russian players are fighting/battling/competing for the gold medal.”

No-one has mention that you forgot the "a" ie a gold medal. Asian speakers often forget the "the" and "a" because I believe that doesn't exist in that language.

First, it’s wrong to use the words ‘game’ and ‘player’. There are several options such as ‘sport’ and ‘competitor’ that are natural. As far as ‘fight’ it is usually unnatural to say they are ‘fighting each other’ unless it is truly a fighting sport like boxing. But you could get away with ‘fight for’ as well as ‘to be in a fight for’ an Olympic medal’. Another option is ‘vying for’ They are fighting for the gold (not golden) medal. The athletes from the two countries are in a fight for the gold. (The first two examples suggest that other competitors don’t have a realistic chance at a gold medal) Both competitors are vying for their first Olympic medal.