Will smart-contract work when the client only finds 1 peer?

What are the downsides to running with only 1 peer in the client?

And what are the downsides for the daemon, if it is able to find 3 peers?

Can I workaround that by just copying a previous peers file that has a lot more peers?

The peers are mainly for double spend protection, but also for propagating transactions faster. It should work fine with 1 peer as long as your full node has enough peers.

Yes, you should be able to copy another peers file, as long as it is not running, or it may overwrite it on shutdown.