Chat rooms are meant to be transitory conversations and as a result their history is kept short. Once a day the system will automatically clean up any messages more than three days old. That is why you have noticed that the messages are disappearing. This behavior is intentional and a functional limitation of this feature.
If you want to keep a history then you will either need to use Chat Events, which obviously would need to be scheduled each day but would leave their histories in tact, or perhaps you could use another feature like Topics where the conversation will be persisted forever.
When you block a person, they can no longer invite you to a private message or post to your profile wall. Replies and comments they make will be collapsed/hidden by default. Finally, you'll never receive email notifications about content they create or likes they designate for your content.
Note: if you proceed, you will no longer be following .