Is it possible for a user to automatically generate ip addresses and fill my database with a ton of spam?
Usually no. As the ISP DHCP leasing mechanisms nowadays try to pin a specific IP to a specific user as long as possible. They would need a zombie botnet of other people's computer to flood your system this way.
Is it possible for a user to spoof the ip address of another user, and gain access to their links.
Generally yes. Whilst IP spoofing has been made somewhat harder, depending on the protocol; you should assume the IP is not secure or consistent. Besides, users themselve often do not have static IPs
Overall, don't use an IP as the sole source of identity if you can think of other methods.
To avoid using a login you could simply cache their identity in a long term browser cookie or Flash LSO. Users that purge out LSOs (a minority) could get more links by "resetting" their identity (much like newspaper soft-paywalls) - but this depends on how important limiting users to 10 links is for your website.