The title talks by itself. Is it possible to import a private key into an offline Bitcoin Core/lightwallet?
Yes. It is possible to import a private key into an offline wallet.
But how do your computer that is not connected to the internet how much BTC there is on it?
It doesn't.
You could copy the blockchain from an online wallet and then your offline wallet could compute the balance as of the time that the blockchain was copied.
or
You could run an online copy of the wallet with bitcoin addresses only (no private keys). This online wallet could calculate balances and allow you to create transactions. Then you could transport the unsigned transaction to the offline wallet to be signed. Then you could transport the signed transaction back to the online wallet to be broadcast. This is how Electrum Offline and Armory Offline work.
If funds are added, will your balance be actualised ?
Only on the online wallet (unless you copy the blockchain to the offline wallet).
If it possible there is no risk of fraud, by adding fake bitcoins to an address and then spend it elsewhere ?
It is not possible to spend fake bitcoins. Bitcoins don't actually exist in a technical sense. There are only transactions. When you "spend", you are actually spending the unspent outputs of previous valid transactions. Therefore, if there isn't a valid previous transaction to spend, then you aren't able to create a valid transaction.