You only need to do federal paperwork for SBS and SBR in Connecticut. Except, when a sub-12" SB firearm is transferred, it needs to be treated as a pistol (DPS paperwork, auth #) in addition to the federal fun.
You can convert your regular rifle or shotgun into an SBS or SBR after the approved f1 is received.
Regarding general legality-
Connecticut law says nothing about SBR legality.
Connecticut law hinges SBS legality on federal legality.
Carry/ Transport
Connecticut has a funky law defining all firearms with barrels under 12" as pistols. We also define all stocked, rifled bore firearms as rifles and stocked smoothe-bore firearms as shotguns. It is possible for somthing to meet multiple criteria.
You CAN NOT carry or transport a <12" SBR, SBS, or AOW without a Permit to Carry. However, if it has a stock on it, you can't carry/ transport it loaded in a motor vehicle or on a public roadway.
You may carry or transport a 12" or greater SB firearm without a carry permit.
AWB
NFA registration does NOT cancel our AWB.
Post-ban Sub-12" SBS and SBR need to be compliant with AWB criteria for rifles (or shotguns, as appropriate) and pistols. Some people don't know, some SOTs will even sell you non-compliant guns. However, you may eventually get yourself fucked if you decide to build something that is not fully compliant. If you want "the works", get a pre-ban.