When you connect to a remote system with PuTTY (using SSH I assume) and you need to transfer a complete file from that system to your local Windows one, just open a separate SCP connection; PuTTY comes with pscp.exe
to transfer files. Depending on your SSH configuration, with multiplexing this may even re-use the same connection, so the overhead should be fairly small.
If you install an X Server (like Cygwin-X) and use ssh -X hostname
to connect, you can launch a remote terminal / GVIM session and have clipboard synchronization with your Windows host, courtesy of the X Server. With PuTTY alone, this is not possible, since it's a sole terminal client without any connection to X.