Actually, sometimes text from draggable content, such as in a URL in a browser, is suddenly draggable all the way to the other computer. But Apple has limited the clipboard usage a lot, in favour of their paid version of ARD (and old tricks to show hidden ARD things no longer seem to work). For a poor man's copy & paste of just text, you could consider synchronising a document using something like Dropbox.
But, as on Windows VNC only uses a password without a username, and as your Mac allows endless attempts to connect, you actually might want to set up VNC to connect through a secure SSH connection. (That is: if your Mac's SSH blocks an account for some time upon subsequent login failures; something to validate.) Once the secure SSH connecting is established, the VNC client would then connect to "localhost", and hence "tunnel" the screen sharing through that secure connection. Once you have that SSH in place, then you can also share that copy & paste document through SSH, without the need for third-parties like Dropbox. And maybe (but I doubt it) that would solve the timeout issues as well.
A few hints to help you search if you want to set up VNC over SSH:
ssh firstname.lastname@example.org -L 5900:127.0.0.1:5900
To circumvent "You cannot share your own computer", only if you run into that (probably only happens when connecting from a Mac to another machine, not what you're doing):
defaults write com.apple.ScreenSharing skipLocalAddressCheck -boolean YES
Note that screen sharing has only been built-in since Tiger. Old tutorials may tell you to install a server like Vine (though some actually still prefer that). I guess good search terms would include
tunnel screen sharing ssh putty windows mac -osxvnc.
(Let us know what you found!)