Medium/medium-hard COCI problems, USACO gold problems, and some relatively easier problems from oj.uz are very helpful for intermediate-level training contests. Wish there existed some vjudge like system to make virtual contests off of these problems without requiring much effort from the trainers. Any suggestions or websites that I may not know of?
Not that I'm aware of. I had similar needs recently and I just recreated the problems on Polygon and made a contest in the CF gym. Since most of the stuff you mentioned has available test cases, it shouldn't take you longer than 10min per problem.
Some COCI problems are already on vjudge (thanks to DMOJ).
Update: USACO and oj.uz problems are also supported on vjudge.