The water supply is for hooking up a washing machine.
Sure, but it's going to fill slower if you do. The washing machine won't care, it doesn't fill by time, it fills by volume.
The narrow pipe may throw off your "warm" setting somewhat. If the hot and cold lines are not equal, they'll flow at a different rate.
The narrow pipe will help your "hot" setting be hotter - if it's on the hot pipe and if it's a long run. You know when you open a faucet and the hot water runs cool for a few seconds? That is the water that is already in the pipe and it has cooled off. On a smaller diameter pipe, there will be less water in the pipe (almost half as much, 3/8 vs 1/2).