If SpinRite is not reading the SMART parameters itself, then potentially lots.
The SMART (Self-Monitoring, Analysis, and Reporting Technology) values are a set of variables tracked by the drive itself concerning many things from general age related counts (time powered up, number of power ups in lifetime, ...), basic health monitoring (number of recoverable errors recovered from, length of spin-up time, number of sectors remapped due to repeated recoverable error, number of reserved blocks remaining for such remapping, current temperature, historic maximum temperature, ...) as well as explicit failure indicators (number of unrecoverable errors encountered, number of failures in past self-tests, ...). Most of these counters/flags have an associated benchmark above/below which the drive starts to consider itself on the way out.
While SpinRite has moved the data off those dodgy sectors and marked them in the filesystem so they do not get used again, the drive does not know about this. All it knows is that is has more unrecoverable errors in its saved state than it is happy with, and presumably other less serious indicators that its condition is bad and/or declining, and when the BIOS reads this it warns you.
The drive knows its own condition better than SpinRite does. I suggest you follow its warning and replaces it ASAP, in case the issue that caused the bad sectors worsens. It might not get any worse (there may have been a very small imperfection in the surface of one of the platters and everything else is fine), but if you have any data on there that you care about can you afford to take the risk?
One caveat: your BIOS may not be reading the SMART indicators correctly, but as you have actually seen bad sectors reported at the OS/application level I doubt this to be the case, so it might be worth grabbing some software to look at them yourself. There are many utilities available to scan and display SMART parameters from your drives - you might even find one specific to your manufacturer (which may include better descriptions of metrics that aren't common/standard - SMART allows for manufacturer/model specific metric to be stored and read) provided on their site.
Yes.
Stop using it. Take the drive out and attempt to back up everything you want. A HDD should not be used once it's started to fail (Once it's failed, it'll start failing more and more and more)
The drive could die now, or it could die in a year's time. There's no real way of knowing, all a SMART result can tell you is how the drive is performing right now. IMO, it's not worth the risk - back it up and replace it.
Best Answer
SSD drives do support SMART, according to this ServerFault post.
USB thumbdrives may use cheaper memory controllers that do not support SMART, however. It depends largely on the manufacturer and their component choices.