Double signature requirement of Breach Remedy Transaction
If we assume "half-signed" implies that both parties sign the transaction, then the Breach Remedy transaction would also include the spend from the other output in the commitment transaction which would then also require Bob's signature. This logically makes sense (but is not explicitly stated in the paper) as it would reduce transaction fees and consolidate UTXOs. But yes, you are correct, there is no need to have both parties sign the Breach Remedy transaction.

The current protocol design actually does away entirely with the Breach Remedy and Revocable Delivery Transactions by using special scripts which enforce the rules that those transactions would enforce by existing. This script uses a relative lock time (with OP_CHECKSEQUENCEVERIFY) to enforce the Revocable Delivery rule. It also uses a revocation key which can be generated by the party performing the revocation after a commitment transaction has been revoked, thus doing away with the Breach Remedy transaction. The output script construction is described here.