A) You cannot determine whether your tamagotchi is a genuine by whether it has connextion/connection written on it. I have a V1 and V2. One doesn't haven't connexion written on it and the other has.
Just because it may have been purchased from ebay does not mean that is won't be genuine.
The best way you can tell if your tamagotchi is genuine is to have a look on the packaging for the little red Bandai logo square.
Be sure to check the package and make sure it says "TAMAGOTCHI" and not something else.
And on Ebay sometimes they have FAKE tamagotchis like V4's and +up so be sure to check the package at all times. Or the problem could be that you are not connecting correctly.
To connect simply go to Ver. 3 if connecting with "VER. 3" or others for V2 and V1 Tamagotchi's. When connecting be sure to press the "A" button to change the word "Stand by" to connect. To ensure it stays on "Connect" hit the A button repetively to ensure it stays on connect if that helps you.
My friend Cara had a fake once, but it always connected, but it was last year and we didn't have the cheats so I don't know if yours is the same but if the answer to this question is YES, it's a faulty.
Q:
Is your V2 a clear blue colour unlike any of your friends?
It would really help if you could post a link to a picture of your tamagotchi up here. I will probly be able to tell you straight away. If that is not possible, then here are some tips:
Look for the Bandai logo on the packaging
Check the screen is the right size and pattern
Look at the keychain. Does it look similar to other tamagotchi?
Check that it is the right shape and size
If your still not sure after doing the above, get another tamagotchi v3 and compare it, right down to the last detail.