Let me first try to paraphrase the meaning of each of the translations you mentioned.
To me this has a connotation of getting supplied by an established supply chain. You are working in a Pizza place, and are requesting some chopped tomatoes now from the person doing the tomatoes. Or you are an emergency physician on site and are requesting a helicopter for one patient. The request is likely anticipated, but the actual need, or perhaps the timing or quantity, are part of the request you communicate.
Anforderung can also denote a requirement, e.g. in a job description, but that's a different meaning.
In my experience rarely used as a noun. The verb abfordern is similar to abverlangen (which I'd consider the more common form) and suggests a demand which might put a considerable burden on the one being asked: “da wird den Schülern ganz schön viel abverlangt.”
This is a request or demand you expect to see satisfied. It's most positive use case is probably eine Aufforderung zum Tanzen, i.e. asking someone to dance. Even there, in many contexts social rules suggest it would be impolite not to honor such a request. In another positive use case, it can mean encouragement, like a teacher encouraging students to try something out.
But this might also express a pretty strong demand, with the possible implication of some legal reaction if not honored. “Meiner Aufforderung, den Hund an die Leine zu nehmen, ist der Beklagte nicht nachgekommen.”
Sounds pretty formal. I'd only use the verb in a formal written letter: “Ich ersuche Sie daher, …”. That letter itself may then be referred to as Ersuchen: “Ich habe mein Ersuchen gestern abgeschickt.” I'd say it's quite polite but still moderately determined.
Rarely used, but sounds mostly like Ersuchen.
This has very strong legal connotations. The offer to enter a legal agreement is an Antrag, so to a lawyer there are Anträge everywhere whenever he goes shopping. To the rest of us, the word Antrag evokes a picture of some form to be filled out in order to file some request with some official institution. “Ich habe den Antrag auf Kindergeld ausgefüllt.”
There is also the case of “Er hat mir einen Antrag gemacht” indicating that he asked her to marry him.
This is often part of a contract. You're asking an artisan to manufature something to your specifications. Or you're asking the garage to repair your car. It is their business to satisfy requests like yours, so this feels pretty neutral.
Looking at your categories now, I'll try to suggest suitable words for each of the cases.
asking for a favor (with degree of friendliness or neutrality),
I'd say that's Bitte, or perhaps Wunsch if you want to convey that it means much to you.
requesting that someone who is responsible for something does that thing (neutral)
Auftrag for a request directed at someone performing their profession.
Antrag if directed at a public office.
demanding that someone who is responsible for something (strong to very strong, or neutral).
Aufforderung is quite determined but polite.
Forderung is even stronger. The word is also used to describe a claim you have e.g. after lending someone money, so this implies even more strongly that you have a right to make such a demand.
(Your question made me think of two songs by Reinhard Mey. One is Ich möchte! which gives examples of very different applications of the verbs mögen, wünschen, fordern. Each can be used in very humble and polite ways as well as very demandin and forceful ones, depending on context. The other is Einen Antrag auf Erteilung eines Antragsformulars. It makes fun of formal requirements when dealing with public administration. The character in that song is trying to obtain an application form (Antrag) required to be granted (Erteilung) a second application form (Antragsformular), applying for a written confirmation (Bestätigung) that some carbon copy (Durchschriftexemplar) has been voided (Nichtigkeit).)