Can someone tell me if this is a valid LAN connection.
A PC is patched(plugged) to a RJ45 box(box 1)
A patch cable, CAT5 is run from that RJ45 box (1)to another RJ45 box(box 2) 20 feet away. Then a short CAT5 patch cable is run from RJ45 box(2) into a fiber converter.
Will this configuration cause signal problems?
More than likely this will not cause any problems. The cable is a short length, and you'll have some small signal losses at the termination points (which is to be expected), I'd expect the greatest hinderance to the signal might occur at the media convertor. However, in my experience I haven't seen a media convertor have much of an effect on signal quality.