GigabitEthernet SFP transceivers are supported in 1/10G SFP+ Nexus 5k slots. Please see the Nexus 5k Datasheet (see Table 2) for details.
Original Answer: No FastEthernet SFPs are listed in the Nexus 5k Datasheet (see Table 2); as such, you're out of luck. FastEthernet SFPs are not supported on the Nexus 5k; however, many FastEthernet switches have GigabitEthernet uplinks. If that FE switch has a GE uplink, then use that GE port on the FE switch.
The old GBIC transceiver format was pretty generic and you could ignore branding on them. SFPs are a different animal, vendors qualify individual part numbers for support; you can even see different transceiver part number support between different switches from the same vendor.
The Cisco WS-C2960S-48-FPS-L has 4 SFP ports which operate at 1Gbps.
From the documentation I can find on the Broadcom 57840S, it has four SFP+ ports that operate at 10Gbps. I find no mention that it can support lower data rates.
As for your Cisco Twinax cables, the only ones I am aware of are 10G. These will not work in your 2960S. A couple of posts that may be somewhat relevant to this from the Cisco forums would be here and here
Best Answer
Assuming you have compatible SFP-T transceivers, Yes.