Wrong: He is six feet in heighth.
Right: He is six feet in height.
This is a simple misspelling/mispronunciation, likely caused by the fact that width and depth each end with th. Height rhymes with night; it ends with a hard t sound not a th.
Inequity vs. iniquity
Wrong: He visited a den of inequity.
Right: He visited a den of iniquity.
Unless there was something unfair about that den, the correct word here is iniquity (sin or wickedness). Inequity refers to unfairness, injustice, or bias.
Alot vs. a lot vs. allot
Wrong: Thanks alot!
Right: Thanks a lot!
This is a simple spelling error. There’s no such word as alot. (And don’t confuse it with allot, which means to “parcel out” or “divvy up.”) If you dole out items frequently, you might allot a lot.