With cut-through switching, forwarding starts right after reception of the MAC destination address. This however may cause broken frames to be forwarded across a switch - simply because the frame integrity can only be checked when looking at the whole frame, from header to FCS.
In this context, fragment free means that a cut-through switch monitors the integrity of each frame while forwarding. When an ingress port exceeds a certain error rate the switch changes over to store-and-forward switching, limiting the overall number of forwarded, broken frames originating from malfunctioning ingress links.
It is because most errors occurs within the first 64 bytes of the frame
That isn't true. Error bits can be located anywhere within the frame. Also, there's no way to detect a damaged frame (not from collision) before receiving the trailing FCS.
64 bytes or 512 bits is the slot time for half-duplex Ethernet. Looking for continuing frames beyond that limit or a lack of collision guarantees that no runts from collisions are forwarded. In obsolete half-duplex mode those runts may be very common, so not forwarding them is a very good idea.
A late collision is a collision happening beyond that slot time, outside the normal collision window. They (used to) happen when the network exceeded Ethernet's design limits (segment length, 5-4-3 rule etc.). You can't always catch them (as they're late) and you can only detect those broken frames by failing FCS check.
If ethernet uses duplex which it mostly does in the modern world, why would there ever be a need to check for collisions?
With full duplex there can be no collisions. However, up to Fast Ethernet speed (100 Mbit/s) most devices still support half-duplex mode (HDX). A cut-through switch with a half-duplex link could theoretically forward fragments from collisions received on the half-duplex port. In practice, most cut-through switches use store-and-forward for HDX source ports (as part of the fragment-free strategy).
does fragment free by any chance relate to IPv4 fragmentation?
No. "Fragment" here refers to incomplete frames (mostly) due to collisions. IP fragmentation is an inefficient, yet necessary and legal mechanism to forward packets over a network not supporting the actual packet size (packet size > MTU).