Can you fake messages from recorded message-history? Lets assume Eve has a lot of PGP conversation with Alice and records all those messages. Bob is also regularly communicating with PGP with Alice. 

If Bob doesn't care about not having a signature from Alice, could Eve send any of those recorded messages unchanged but without signature to Bob pretending it came from Alice?
If Bob does NOT care to check signatures (as in the question), Eve can send ANY message she wants to Bob pretending to be Alice, including but not limited to messages Eve got from Alice; all Eve needs is Bob's public key (which, as the name implies, is assumed public knowledge thus known to Eve) and straight use of PGP.

Therefore the right question is: Can Eve reuse signatures she got from Alice, so that Bob concludes that a message sent by Eve is from Alice, even though Bob DOES care to check signatures?

The answer is YES, if Eve sends to Bob a message with exactly the content the message had when signed by Alice. So if Alice sent to Eve a message like "Let's meet at the bar at 11 pm", Eve can impersonate Alice and send that same message to Bob, who might conclude he has an invitation from Alice, when that invitation really was for Eve, perhaps another bar than the one he expects, and/or a past day. PGP gives insurance to Bob that the message was signed by Alice, and gives him a chance to detect a change of day (for PGP signatures include a "signature time stamp"). PGP does NOT give insurance to Bob that the message was for him.

Note: having an history of many recorded messages only helps Eve by giving her a wider choice of messages she can send to Bob pretending they came from Alice; Eve can not cut and paste fragments from past messages to compose a new one.