I'm going to take a markedly-different approach from many of the others; hopefully this perspective is useful, too.

Start by looking at the nearest-neighbor spell to augury: aura of life. 30' radius, resistance to necrotic damage, regains 1hp at start of 0hp turn. Very easy, and the only thing really left up to interpretation is "nonhostile."

Most spells are like this: they're very clearly... spelled out, using defined terms. [/rimshot]

Augury is a different type of spell.

Augury is a super-vague spell, with lots of room for GM interpretation, preference, style, &c. And you're a first-time GM, so you feel like you've got little guidance from the spell. You're right, by the way.

There's a reason for that: while most spells are a manifestation of the player character's will, this spell is designed to do something fundamentally different. Augury allows the player character to engage in conversation with the GM. Not the player; the player character.

I want to be clear on this: the purpose of augury (and some other divinations) is markedly different than most other spells. Once you realize that, things fall into place.

At some tables it's perfectly appropriate for a GM to say to the players "hey guys, if you take on this dragon you're all probably going to die." If necessary, this is justified in-fiction as "your characters are noting the number of dried skeletons of previous adventurers, they know stories of Smaug from their childhood, they've trekked through miles of desolate wasteland just to get here." Or it's hand-waved away.

At other tables that would be completely unacceptable. It'd ruin immersion, it'd destroy people's roleplay, it'd cheapen the game. Crossing the boundary between in-world and at-table knowledge/conversation/interaction is forbidden.

Augury explicitly crosses that boundary, in a way supported in the fiction. Augury allows the character to talk (specify course of action taken very soon) and the GM to respond: "that should go well," "uh, I wouldn't do that," "kinda mixed bag," or "meh."

But what about timing, subject, and adjudicator?

My best advice is never to spend more than 30 seconds on an augury. Listen to the question/proposition, lean back, think for five seconds, and give an answer. If your players or PCs want better answers, they can wait (or pay) for divination, commune, or foresight.

What does the "30 minutes" exactly apply to? Initiating the course of action, since 'taking a course of action' is not necessarily 'completing' a course of action.

Who is the subject of the result? Typically the caster (since the range is 'Self'), which normally also encompasses people the caster wishes well (like his party or innocent bystanders). The player however is not a relevant entity in the game world (bad things for the PC would appear as woe even if the player might enjoy them).

Regarding "The party wants to visit town, but they are worried the Rogue might be recognized and jailed": the party might expect 'woe' to be mentioned if that was the case. The caster may not care about the rogue, but the rest of the party does, and presumably the caster cares about the rest of the party.

Who judges whether the result is good or bad? The DM (with the caster in mind). The "otherworldly entity" is not necessarily the Cleric's deity.

A. Timing

The action is what is specified within the next 30 minutes, not the results. So the results need not occur before then.

However, I see nothing in the spell text to suggest that the action must or may not be completed. However, what is considered the action, and what is considered the result depends on your point of view. In your case 3, receiving the staff could be considered the results of your signing the contract. Thus the players should receive Weal, provided they sign the contract within 30 minutes.

I think that it makes more sense, given the flavor of the spell, to say that if they started on the action within 30 minutes, that it should qualify as "taken", and would rule it that way in my games. But I think it could be interpreted either way. If you want to make a magical ritual that takes 35 minutes to complete because the players won't be able to rely on the spell (maybe that's why the contract takes 35 minutes to complete?), then go for it.

B. Subject

The spell only says you so it implies the caster only. However, if it were cast as a ritual, and all the players helped, I would say that it would be the collective you, meaning the entire party. But that would be homebrew.

Regardless, this is the sort of thing you should tell your players, since, regardless of which way you choose, it's the sort of thing that someone who regularly casts that spell would know.

C. Perspective

The spell does not specify that the otherworldly entity must be associated with the caster or the party in any way, or that the players can choose who to ask. This would be a perfectly acceptable homebrew modification to this spell, but then it would be homebrew, and outside the expertise of RPG.SE

As far as who judges benefit/harm, I think this one has to be the caster. There would be little purpose in casting a spell to ask a random entity if they should take that action, since there is a non-zero chance that the entity hold opposing goals as the party. This is purely based on the fact that the spell would be next to useless without this clarification.