Crowds of People Under Alien Influence: Nope, can't make this one work even at a stretch.
Tie-in with Doctor Who story: Again, more plot-ripoff than actual tie-in, with the traditional quest-through-time-and-space format (q.v. The Keys of Marinus, The Chase, The Key to Time) heavily exploited. And the idea of Emily's granddaughter giving the key to Sarah Jane at the right moment is a steal from "Blink."
Rani's Mum is Annoying/Is Absent: The latter.
Luke/K9 Cameo: Not even a mention. How soon we forget.
Sarah Jane Waxes Maudlin: In episode 2, when telling Emily how lucky she is someone loves her.
Mobile Phone as Plot Device: Clyde determines that he's in the past by checking the signal, and Emily gets in a (ahem) heavily telegraphed comedy moment as she marvels at the idea that "Mr Bell's invention" not only caught on, but went wireless. Plus it seems MP3s have Nazi-repelling capabilities.
Racism Towards Aliens: Clyde, of all people, tells an SS officer that he's a blind bully who judges others only on appearances, and hates and fears anyone who's different to him. Tu quoque, Clyde.
The Crimes of Sarah Jane: None, though Rani gets in one count of Grand Theft Music-Box, and another of Impersonating a Tudor Personage. And the characterisation of the pantomime Nazis is indeed criminal.
Sonic Lipstick: No.
Wristwatch Scanner: Used to scan for "ghosts."
One or More of Sarah's Companions Falling Under Alien Influence: Well, everyone goes into the past under alien influence, technically.
Sarah And/Or Companion Acts like a Selfish Cow: Actually, Rani gets to be nicely unselfish for a change, giving up the chance to go back in order to urge Queen Jane to martyrdom.