... can others on the same IP address log into your accounts?
Authentication has nothing to do with the source IP address and thus with a shared use of a VPN - unless the authentication is explicitly done by source IP. For the common use cases you describe (Facebook login etc) the authentication is not done by IP address but by username and password and after that bound to a session cookie which itself is stored only in the specific browser.
This means that users sharing the same browser would be treated as the same but not users using the same VPN or otherwise using the same IP address (like all users behind the some home router). Even different users on the same computer would be different if they are using different accounts and thus different browser profiles.
... Can someone on the same VPN network tap your traffic and cookies, gaining access to your private accounts?
Sniffing the traffic is a different scenario from just sharing the IP address. For one, the VPN provider itself could of course sniff all the traffic. Usually sniffing between users in the VPN is not possible but a misconfiguration of the VPN could actually make this possible. Apart from that a VPN only provides protection from the local VPN entry to the VPN exit. Since the VPN exit is almost never the ultimate target attackers could sniff the traffic after the VPN exit too. Therefore you should use end-to-end encryption (i.e. HTTPS) to the target no matter if you use a VPN or not.