Do I need to secure an automatic generated private key used for locking an anchor transaction (the one that opens the channel, locking the client funds), since it only serves that locking purpose (not holding money)?
Can I just drop it? Any further advice? Thanks.
If by "anchor transaction" you mean the transaction that locks some amount of bitcoins to a multisignature address from both yours and your partners private key, then you must keep your private key safe because you will have to sign transactions in both the cases when you will pay, or if you need to refund all the bitcoins back to yourself after some time has passed.
After reading you answer again it seems as if you are looking from the perspective of the receiver. In that case you will still need the private key to be able to sign the (last) transaction that sends you bitcoins from the multisignature address.
