Applause actually sold over 600k first week but Billboard was tired of her slaying so they posted those fake first week numbers. They planned Gaga's flop era.
Wide Awake. Pretty sure that song was supposed to hit #1. Billboard just wanted to give Carly more weeks at the top to keep whatever hype she had going (which, obviously, didn't sustain itself after CMM died).
I think the most obvious is See You Again + Uptown Funk being #1 for as long as they did. Bad Blood was literally above SYA in every single format and SYA was still #1.