regarding no1 - There is no Heavycoin windows installer, so I think something needs to be configured in your Windows to make the association work.
Gotcha - figured that part might have been handled in the client itself

However, that doesn't explain the error thrown when feeding the client the URI manually, e.g.:
heavycoin-qt.exe heavycoin:HTQUFnH79dorHhfJATgLDUAwNpQ2Xa9LLb
or
heavycoin-qt.exe heavycoin://HTQUFnH79dorHhfJATgLDUAwNpQ2Xa9LLb
yields:

I'm not sure this is all there is to it, but I'd guess that at least this one line was overlooked:
https://github.com/heavycoin/heavycoin/blob/master/src/qt/guiutil.cpp#L83I've fixed heavycoin:// URI parsing issue and tested it this time / plus updated the builds, sha256 checksums, and signatures. [It was what you found, but also the length of a string replacement for stripping out "//" needed fixing too.]
If you download 1.3 again you'll get the patched version (1.3-4-g565b5a9). This build includes improvement to the Chinese translation of "Heavycoin".
The windows installer will have to wait for the next release.
Thanks for your contributions