Phrase Meaning: He Inspires Great Loyalty in His Staff – Superior or Coworker?

The word former refers to state. It means that the person or object was something, but no longer is. The word "former" only refers to sequence in special cases where it is contrasted with the word "latter" (in which it means the first of two given items).

The word previous refers to sequence. It means that the person or object was/did something before something else took over or replaced it. As StoneyB rightly clarifies in his comment, "the previous" means the directly preceding member in a series, however "a previous" can refer to any earlier member in the series.

The word last, in this context, means the same as "previous", but is only used to refer to the immediately preceding item. (Jay warns that "last" can be ambiguous as it is also used to refer to the final entry in a series.)

Thus, you can say "former coworker" of someone who is no longer a coworker. However, the word "previous" means the one before the current, so saying "previous coworker" or "last coworker" does not make sense unless you had a coworker and they were replaced.

Referring to TV series, you can say that you preferred the "previous" or "last" show, and this would refer to the show immediately preceding the current show. You cannot say the "former" show in this context.

When referring to a show that aired earlier than the immediate previous, you can say "a previous show". Otherwise, you can be more specific. Depending on the circumstances, you might say "the first show", or "one of the earlier shows", or "episode ten", or simply "one of the older shows". You could even say something like "five episodes earlier".

I have to disagree. Grammatically, the sentence is doing no more and no less than asking to what extent the other person likes the shirt.

Compare: How do you like your boss?

This does not imply any favor or disfavor on the part of the person asking.

That said, practically speaking, when you ask someone about a shirt, you are typically considering buying/wearing it, and thus typically have at least a semi-positive opinion of it. But that's true of all your example sentences.