NO as long as they are involved in the activity that their commercial license permits. The finfish/lobster licenses have their own reporting requirements attached to them. The Federal Registry was intended for recreational fishermen, not commercial.

For that, Yes. Their lobster license permits them to fish for lobster. If they want to throw a line in for finfish while they are hauling pots they need the recreational marine license.

If they are just lobstering, their lobster permit covers that. If they have a commercial finfish license and are using nets, or whatever gear their permit allows they are good to go to catch finfish with that gear. If they want to fish for recreation purposes (i.e. with a pole and reel) they would not be covered by their commercial license unless their commercial license allows them to use a pole and reel.

Sec. 26-142a. Commercial fishing vessel permits. Registration of nets and areas of use. Registration of charter boats. Fishing licenses and egistrations. Possession limits. Fees.
(a)No person shall operate, use or attempt to operate or use a vessel for commercial fishing or landing activities authorized by this section unless the commissioner has issued a vessel permit for such vessel to the owner of the vessel. No person shall use or assist in using commercial fishing gear in any water of the state or land in this state any species taken by commercial fishing gear or for commercial purposes, regardless of where such species was taken, unless such person has been licensed by the Commissioner of Environmental Protection to use such commercial fishing gear or land such species; except that any person who holds a license to set or tend gill nets, a license to take lobsters or fish for personal use, a resident commercial fishing license, a nonresident commercial fishing license or a commercial landing license may be accompanied and assisted by persons not so licensed. A resident of a state which does not issue commercial licenses to take eels to residents of Connecticut shall not be eligible to obtain a commercial license to take eels in the waters of this state or to land eels in this state. No vessel shall be used to land any finfish, lobsters, crabs, including blue crabs, sea scallops, squid or bait species for sale, barter, exchange, consignment or transportation to any point of sale unless an operator of the vessel is licensed for such purpose, except that any person who holds a commercial fishing license issued by the commissioner to fish by the method used to take such species, regardless of where such species were taken, shall not be required to obtain a landing license. No person shall take or attempt to take lobsters for personal use by hand or by scuba diving or skin diving unless such person has been licensed by the commissioner to take lobsters by such methods. No person shall take or attempt to take finfish for commercial purposes by the use of hook and line, including but not limited to rod and reel, hand line, set line, long line, or similar device unless such person has been licensed by the commissioner to use such gear for commercial purposes, except that notwithstanding the issuance of such a license, no person shall take finfish for commercial purposes in the inland district by the use of hook and line. The use of a purse seine or similar device to take species is prohibited.