Again WOW, WOW, WOW...

I could make LimeWire Pirate Edition (LPE) 5.6.2 to work again: set as 32-bit mode and replacing its
/Users/~/Library/Preferences/Limewire/gnutella.net
by the one of WireShare at
/Users/~/Library/Preferences/WireShare/gnutella.net

Thanks to Lord of the Rings for the tip about the gnutella.net file- Although deleting it was not enough. What worked was replacing it by the one from the working WireShare. Amazing!

It shows a blue five-bar turbo-charged connected to five hosts. Shows Java 1.6.0_65. On the other hand, as previously indicated, LimeWire - Preferences - Advanced
is set as default:
Gnutella port - Use Universal Plug 'n' Play (Recommended)

BitTorrent ports - Use Universal Plug 'n' Play (Recommended)

Of course, the Mac and Little Snitch firewall have permission for LimeWire to connect.

So now, both LimeWire and WireShare work the Mac mini with OS X 10.9.4 (Mavericks).

Just for the record, in case it may help someone.