DWV insert pipe into run
Rubber adapter couplings are what is used for what you want to do. You will need to insure that you order the correct one as copper and PVC have different outside diameters.

I do not think that you will get good heat transfer by simply wrapping soft copper around the drain line as there will be minimum contact. There are commercially available products on the market that do what you are looking for that will provide much better heat transfer. Here is one such product: Recoh-vert.

You'll notice that the pipe-in pipe method archives much greater contact between the hot drain water and the incoming cold water. They also install these on the vertical as what travels around the inside wall of the pipe when draining vertical (as opposed to just the bottom when draining horizontally). This also increases the amount of heat transfer.

You can make a similar product yourself, though I question the possibility that a home made product would prevent cross-contamination. Defiantly would not pass any local codes. Here's a video on how to make one: Shower Heat Exchanger.

If I understand your situation, you have limited space in install the heat exchanger. I believe that there are also p-trap versions available.