Answers:
Is 'Respektiere' in the imperative?
Yes, sort of, at least that is what it means. It is the Jussiv, a replacement form for imperative that uses Konjunktiv 1 as the verb form.
Jussiv is a bit archaic nowadays and not very often used, at least not in everyday conversation. Imagine the King asking his servant to serve wine, he would have said:
"Schenke er"Schenke er mir Wein ein"
Which is the same construct, or a typical excerpt from a cooking recipe that tend to start with "Man nehme ..."
"Man nehme eine Flasche Wein und schütte sie in den Koch"
What case is 'jeder' in (e.g. nominative, dative)?
Nominative. Pull it in front to see why: "Jeder respektiere...". Also see next answer why it needs to be.
How does 'jeder' contribute to the sentence?
It is the subject of the sentence.
What does the whole sentence mean?
While the jussive mode is typically expressed in English with something like "let's", for this specific example I would rather choose a form of "to be to":
Everybody is to respect the situation he is in: This will help everyone.