Angel (possessed by Cordelia) writes an address on a wall. The address is gone in the next shot when he's blown backward.
In Angel Episode 1.15 The Prodigal, Angel Investigation gets their security system, Cordelia tells Angel that the code should be something simple, "Like my Birthday" Angel tells Cordy that he doesn't know her birthday, and she replies that after punching in 05-22 for "11 1/2 months", he will definitely remember her birthday. In Season 3 However, there is an episode set on Cordelia's birthday, and that occurs in January, not May.
When Wesley is bent over after having stabbed the demon (at around 1 min) his left arm is visible beneath his shirt, bent behind his back.
Angel demands that Cordelia wash her hands before holding Connor, but doesn't seem to mind her holding Connor next to her very dirty clothes.
After becoming part demon, Cordelia feels no pain at all while she's having a vision, yet Doyle was half-demon and suffered excruciating pain when he had a vision.
When Cordelia is in her alternate reality the Hyperion Hotel is still in business. Her alternate reality has nothing to do with the hotel, it went out of business in the 50's, long before she was born so her reality change wouldn't have changed the fate of the hotel.