I recently moved into my grandpa's old farmhouse and it is a mess - almost nothing is working like it should:( Now I've noticed some plumbing problems... I wash my clothes in warm wash/cold rinse. Realized the other day that there's just a trickle of cold water coming out for the rinse water (no wonder it was taking so long to wash! - lol) I've turned the cold water all the way open, so it should be coming out--- what's going on?

From the experience I have with the country water we have,this could be:
1) The cold water line (where it goes into the washer) is clogged with debris/sediment
2) The actual valve where the 2 water lines go into the machine is clogged for the same reason.
Try taking the waterline off (cold) and see if the screen(should have a screen on the end of the hose or built in to the machines mount) is blocked,if not ask appliance store for replacement fill valve (not too hard if your handy