I'm afraid I don't quite follow what's going on?

Rin is checking the their interactions and not detecting any competition until the last panel where there's disapproval.

And "Propose" is from the anime; in Ranko's episode, where she's trying to talk to the Producer about how she wants her debut to go, she keeps trying to say "Producer!" and wimping out into a different word starting with "pro-", which P-kun is dutifully writing down in an attempt to understand her.

(The punchline is her running out of words, and dejectedly walking away after lamely coming up with "purin" (pudding).