How to prevent Safari from stopping downloads when sleeping

Since the Screen Sharing feature seems to be the only one unable to connect, the best way to diagnose its behavior is by using the application directly.

Please, launch screen sharing located in:

/System/Library/CoreServices/ScreenSharing.app

The application has no visible "UI" so don't expect a fancy Cocoa interface. Instead, once you have loaded it, all you'll see is this "poor" icon in the Dock:

Click on it and then go to Preferences in the Mac menu bar. You should be presented with:

Notice the second option to only encrypt passwords and keystrokes. That's fine, but as you can see, there's no option to skip encryption altogether. Some VNC versions have incompatibilities with encryption, but this would be rather shocking since you're talking about OS X vs OS X.

In any case, once you have the settings like that, try connecting, again, in the menu bar: Connection -> New (or ⌘cmd + N).

A simple dialog appears asking you to enter the Host's name. Following your screenshots, try: 192.168.1.99 and hit enter.

We'll see what happens from here. Also, have the Console.app open on both computers, to see if there are any System messages going on (or anything related).

This was eligible for me: edit /Library/Preferences/SystemConfiguration/com.apple.Boot.plist like this:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>Kernel Flags</key>
    <string>darkwake=0</string>
</dict>
</plist>

and restart your mac. Mountain Lion forces me to enter password twice but it wakes up and works.