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 contact other plane.
First, answering the question as posed: No, there are no rules (actually guidelines) about it, at least I couldn't find any in either the DMG or XGtE, where most of the general guidelines are found.
This is not a problem, though, because when you are DMing, you have two major options: Either you are running a premade campaign or you are running a homebrew campaign. In both cases, you problem can be solved.
Premade Campaign
Most published adventures state the magic item itself, rarely telling you to roll for it. Usually you won't have a problem there. If you do, handle it the same as the homebrew campaign below.
Homebrew Campaign
This is your world. When you wrote this:
Just making the decision myself about what they find, however, feels quite arbitrary.
IMO this is just wrong. You are the world designer. Deciding what monster will be there, how they behave and what treasures the PCs will be rewarded is part of your job, or, if you prefer, is part of your powers. You might think it's arbitrary if you don't put too much thought into it, though, which leads us to...
Give life to the magic items
The DMG has some suggestions on how to create magic items. Check p. 141 for my point.
You can add distinctiveness to a magic item by thinking
about its backstory. Who made the item? Is anything
unusual about its construction? Why was it made, and
how was it originally used? What minor magical quirks
set it apart from other items of its kind? Answering
these questions can help turn a generic magic item,
such as a +llongsword, into a more flavorful discovery.
But don't stop there: Why is the item here? Why is that +1 Longsword, that now we know it's actually the sword of Percival, the Holy Knight of the Kingdom, which was blessed by the High Priest 300 years ago, lost in a freaking Orc Cave? Did they raid Camelot recently and got it as a spoil? The feeling of "arbitrary decision of putting it here" will be nothing to you or your players when you can put a background there, when the item has a plot and simply makes sense for it to be there. That lore can be created before the characters, so you are actually unbiased and not making a world tailored for them.
Obviously be careful with this line so you don't give them too many useless items (for them). But this shouldn't happen in a fairly balanced party - someone will be able to use that staff, dagger and sword.
Best Answer
5e can't handle this question very well
This is, in essence, the same question as whether a player should get circumstantial advantage, disadvantage, or neither or a roll. Taken literally, almost every situation should receive an answer of "weal and woe", just like every roll should be made with both advantage and disadvantage due to the myriad tiny influences in both directions. The game doesn't work well when run that way; I've tried.
Ignoring this question usually works okay in practice
Instead, you have to turn off your critical thinking brain and just kinda go with whatever seems more socially acceptable. Some groups expect tactical advantages and won't bat an eye at being given advantage or disadvantage on an attack roll for swordfighting on the favorable/unfavorable side of a slope. Some groups expect GM-given advantage/disadvantage to be synonymous with "that's cool!/lame!" from the GM and will be confused when they get disadvantage for trying to assault opponents on a spiral staircase.
The factors that are important to a group when they assess your decision to respond with "advantage" or "woe" or similar will vary from group to group, and if you think too much about the rules-text and DMG advice instead of trying to please the crowd, the system breaks down and no one's happy. Instead, think about what your players will think made sense in hindsight, then give the people what they want.