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).
It sounds like you need a client management solution. I prefer the Casper Suite, but in the interest of full disclosure, I work for the manufacturer. However, I was a customer for some time before becoming an employee. There are competing products and I advise you to evaluate more than one solution before making a purchase.
With all that said, a client management solution will give you a central inventory system, and centralized management tools. The information regarding your inventory and management tasks are stored on a server that is accessible to your entire staff, and depending on the privileges you assign, all members of the team can perform management tasks.
I hope this is helpful.
Best Answer
A VNC server on an un-jailbroken device is impossible due to Apple restrictions. Any such app would not be able to run in the background (without a jailbreak tweak to modify the backgrounding code) and is not permitted by App Store regulations anyway.