Can the Lightning Network work and live independently from Bitcoin? Is the Lightning Network Protocol capable to serve micropayment services without broadcast any transaction to the blockchain?
No, not without any transactions. The Lightning Network is based on so called channels between 2 users. 

The creation of this channel has to be broadcasted to the blockchain (and another transaction is needed to close it again).

While creating the channel, you've also to put some BTC into the channel, which then can be transferred between both ends.

To have an example here:

A and B are creating a channel between each other. A creates the channel and puts 1 BTC into it, B doesn't add any BTC to it. This creation requires a transaction on the regular BTC blockchain.

Once the channel exists, A sends 0.5 BTC to B (maximum is 1 BTC as this was the ammount the channel was created with). A few days later B sends 0.1 back to A and closes the channel.

Now the transaction to close the channel will contain: Transfer 0.4 BTC from A to B (as this is the final balance), and only this transaction will be executed. 

No matter how many transactions will be performed between A and B, only the final balance will be mined into the chain.

But that's not the only trick with the lightning network. If A has a channel with B and B another channel with C, A can also send BTC to C (over B) without the need to have a separate channel between A and C, but only the minimum BTC of channel A->B and B->C is transferrable. So if A -> B has a cannel value of 500 BTC, but B->C has only 1 BTC volume, A can only send a max. of 1 BTC from A to C.

While transactions within those channels are free of any transactions on the blockchain, opening and closing will require a transaction. Closing is also required to receive the BTCs initially "stored" in the channel.

Hope this clarifies a little bit how the LN technically works