Will I have warm/hot water in the cold side? If so, how long does it normally take to flush out?

Yes, the water in the cold supply at the check valve will be slightly warm. It only takes a few seconds for the warm water to flush out and get cold. My house has the plumbing in a concrete slab, it takes 3-4 minutes to get warm water to the far sink without the pump ON, with the pump ON, we get hot water within 5 seconds. You might have different results.

Yes, when forcing the hot water into the cold side the cold will now be room temp or luke warm.