Yes, and you can do it in such a way that your paper wallet stays "cold" if you wanted.  You would need to import the paper wallet's private key into an offline computer or a hardware wallet that you trusted, have the device sign the appropriate claim string, and then report back with the signature.  You would then broadcast this signature to the spin-off network either using the spin-off client, or a tx-push web service.  

If this made you at all uncomfortable, you could also move your real bitcoins to a new (cold address) before you make your claim, for then you know that the private key that claims the spin-off no longer has a real bitcoin balance associated with it.  


It would depend on the details of the spin-off launch.  There are arguments both for and against having a time-limit for making your claim (provided it is sufficiently long so as not to appear as a scam).  In either case, these details would be known at launch.  If the time limit is (e.g.) one year, you could choose to do nothing for 11 months and if the coin is dead just keep on doing nothing.  If the coin is doing something important, then you'll surely hear about it and probably decide to make your claim.