Is it wrong that that game sounded fun as hell to me?
Yes, but in the past they’ve read directly from the roommate agreement and it says things like, “If either roommate is to discover a way to time travel…” etc.
The roommate agreement is a continuity nightmare. It states no pets except service animals, yet Sheldon has had cats and fish; they didn’t bother with any rules about girlfriends because having one was so unlikely, yet there are multiple examples of it having rules about girlfriends; it contains a rule about the apartment flag - with the flag only being seen in a flashback episode.
The Sheldonesque way to explain all the inconsistencies is to posit that each episode takes place in a different universe. Of which there are an infinite amount.
As to the end bit of this episode, it’s likely that the gang told Sheldon that’s how the ‘Cats $20’ thing goes. He has little understanding of how mundane things work and they’re always doing stuff like that to him (and sometimes Raj).
Examples: When going to help Howard move, they got fed up with Sheldon’s disbelief that the normal payment for doing that favour was just pizza so they told him it was a choice between a PS3 or a mountain bike. With Raj, well they convinced him that the Rocky mountains were in Philadelphia.
I’m watching the Barenaked Ladies singing the theme song on an eTalk special right now - cool, mang. 
Jim Parsons, Kunal Nayyar and Simon Helberg are guests answering questions as well.
It’s likely that the agreement strongly favours Sheldon, but also that Leonard is a much more laidback person, so he’s not as likely to bust out the agreement all the time.
I dunno—I’ll go so far as to admit that I once had a cat named Linus Pauling (seriously), but the notion of stroking my inner geek by play Klingon Boggle or Counterfactuals gives me paws.
It’s likely Sheldon proposes countless addendums to the agreement and Leonard, being the agreeable sort, just agrees to them.
My DVR had cut the epilogue off.
Question: could it be read that the whole break up was a con by Sheldon and Amy to get Leonard to not continue to urge keeping her away? They come to the lunch table the day after Leonard asks that she not hang around so much because she is a bit condescending, she acts condescending to Sheldon, they terminate the relationship in front of the friends, who are soon working to get the couple back together and who now will never dare complain about her to Sheldon again.
I sense a Bazinga.
I think the joke is totally about logic, especially since Sheldon takes everything literally. He would interpret “Cats $20” as meaning two things that they are giving away, like “Food Water” might mean, following a disaster.