I don't know if they are hard, but they are things that came up in my first few sessions. They are not necessarily questions that have a single right answer, but it can help to think about them:

• How do you deal with players wanting to be stealthy, especially in combat? The rogue wants to hide and sneak up on the enemy. What do you ask them to roll? What do they roll against? How long will they remain hidden? How can they be revealed?
• "How much does it cost to buy horses?"
• "How fast would we be if we traveled by horse instead of by foot?"
• What to roll if someone tries to grapple?
• If someone has the "frightened" condition, when does it end?
• Your spellcaster wants to move up to another player, then cast "Cure wounds" as an action and then "Healing word" as a bonus action. What's wrong with that?
• A PC is talking to an NPC. The PC suddenly decides to stab the NPC mid conversation, but you are currently out of combat. How do you start combat? What happens if the NPC rolls higher on the initiative order than the PC?
• "Can I roll an insight check to see if he is lying?"
• "Can I take the dead enemy's weapon and armor?"
• Your players realize that 8 hours of rest can heal them up completely. They just fought their first encounter and are low on health. They decide to make a camp and do a long rest. Once they get up, they move on, and have another encounter. Being low on health again, they decide it's best to do a long rest again. This is called the "5 minute adventuring day". How do you mitigate this?

Quote from axpresbom >>

So I'm essentially 3 days away from DMing my first 5e session (and first self-written adventure). I've read both the DMG, MM, and PHB from cover to cover twice. Ask me hard questions to help me make sure I actually know my stuff well enough to deal with stuff on the fly?

Here are some questions so hard you should assume no one on this forum knows the answer.

1. First lesson:  sometimes WOTC will give you two RAW answers to the same question that can't both be true at the same time, and you have to pick one with minimal guidance as to how to make the choice.
   • When someone throws a longsword, which ability score modifier applies to the attack roll, and why?  PHB pages 14 and 194 give two different answers and it's never been errataed.
2. Second lesson:  sometimes WOTC won't give you a RAW answer and you have to make up your own.
   •  A Warlock casts Hex on a target and a Ranger casts Hunter's Mark on a target.  Both of them are multiclassed into Rogue 1 and have the Sharpshooter feat.  They each hit the target while Unseen with a net.  How much damage does each character deal to the target, and of what types?
3. Third lesson:  sometimes WOTC will give you clear RAW that doesn't make any sense to you.  This one does have answers people on this forum can authoritatively give you, they're just weird answers.  Also it's multiple questions, to emphasize the issue.
   • Setup:  Adam and Adam Jr are standing in brightly lit, normal spaces.  Bob is standing in patchy fog, counting as lightly obscured.  Bob Jr.  is standing in dim light, counting as lightly obscured.  Carl is standing in opaque fog, counting as heavily obscured.  Carl Jr. is standing in darkness, counting as heavily obscured.  No-one has any special rules, including darkvision.  Bob tries to look at Adam across Carl's space.  How hard is it?
     1. Does your answer change if we change Carl to Carl Jr?  Bob Jr?  Adam Jr?
     2. Does your answer change if we change Bob to Bob Jr?
     3. Does your answer change if we change Bob to Bob Jr and Carl to Carl Jr or Adam Jr?
     4. Same set of questions but now we start swapping out Adam.

     5. Darkness and fog follow the same rules on the tabletop, even though in the real world they behave completely differently from each other, in ways that will impact gameplay (and potentially immersion).  Dim light/patchy fog and darkness/heavy fog both have rules that flatly impair anyone in them, so Carl and Carl Jr. both can't see Adam or Bob, period, even though that's fundamentally not how darkness works in the real world.  Bob and Bob Jr are at disadvantage to see Adam unless they have to look across Carl or Carl Jr (see below), for the same reason.

        Meanwhile, being lightly obscured has no rules text beyond impairing those in it, which means Adam and Adam Jr are not at disadvantage to see Bob or Bob Jr, even though your intuition probably tells you Bob and Bob Jr should be hard to see.  This is because being lightly obscured simply has no rules for obscuring things in it, only for obscuring the things someone in it tries to see.

        Being heavily obscured has more text which is subject to interpretation: a heavily obscured area "blocks vision entirely".  This is commonly interpreted to mean that these areas work like your intuition says heavy fog works like (but not total darkness): No-one can see Carl or Carl Jr, and no-one can see Adam if they have to look across Carl or Carl Jr.

4. Fourth lesson: lessons 1-3 can combine with each other.
   • Adam casts Magic Stone on a rock, then hands it to Bob, who fires the rock from a +1 sling.  Bob is wearing an attuned +1 Amulet of the Devout, which is like a +1 Rod of the Pact Keeper in case you can't see it (Amulets of the Devout simply have easier to read grammar for the relevant text here).
     1. Is this an attack made with a weapon?
     2. Is this an attack made with a ranged weapon?
     3. Is this a weapon attack?
     4. Is this a ranged attack?
     5. Is this a ranged weapon attack?
     6. Is this a spell attack?
     7. Is this a ranged spell attack?
     8. How do you calculate the attack roll of Bob's attack?  Specify ability modifiers both by which ability and by whose ability, proficiency bonuses by whose proficiency bonus, and any other bonuses.

I can rattle off more examples, but many of them may be arbitrarily picayune and unlikely to come up in your game.  The general lesson I'm trying to emphasize to you is that you need to be ready and prepared to homebrew on the fly - 5E is unplayable RAW because of the issues above being extremely common, meaning there's no one universal way to play.  Absolutely everyone homebrews to some extent at their table, and you need to be ready to make decisions as they come up.

Rules questions are boring. Here's the real stumpers.

What's this guy's name? Yes, this guy on the street I asked for directions and I'll never see again. What's his name?

What street are we on?

What kind of fish are in that lake?

Who's the mayor in the next town over?

What kind of wood is my bow made of?

Will this burn? I mean, burn really slowly, so we have time to get away, or does it go up really quickly so we'll be caught in the fire ourselves?

In line with the others, here is a different perspective:

(1) When in doubt, rule intuitively, but consistently. Oftentimes game momentum is more important than doing everything perfectly. (You can look up the official rules between sessions, so just make a note of anything you improvise.)

(2) Don't be afraid to let your players remind you of rules. If the player suggests something reasonable, just roll with it and look it up later. (DM always has the last word, so listen thoughtfully, then facilitate decisively.)

(3) Asking the players to roll a d20+Modifier is a good way to resolve pretty much anything that isn't well defined. Engage with the players and reward them for clever participation by letting them roll with advantage, and compromise with questionable requests by allowing them to roll with disadvantage.

The bottom line is having fun and telling a story with your friends. No one can be expected to know all of the rules and content in the books, and a significant portion of it will never actually come into play. For everything that comes up regularly, you can make a cheat sheet for the inside of your DM screen.

IMO, the “hard” questions have little to do with anything in the rule books. IMO, the “hard questions” are more like these:

• Why did that little girl do that thing?
• Is there another tavern in town?
• Is there anyone in town who can help [PC X] with [Problem Y]?
• What are you gonna do when the players get stuck?
• Why should those PCs care about that Adventure Point/NPC/Story Hook/Plot Twist?

Say I have a ship. If I replaced one of its parts, is it still the same ship? Let's assume I keep replacing parts until its all new parts. Is it the same ship, and if not, when did it stop being the same ship? Perhaps I put all the old parts together and make a second ship. Which one is the original ship, if either?

The question that has saved my life as a DM: Would this ruling be easily overpowered if I allowed all the bad guys to do the exact same thing? 

This helps me make rulings that are fair and usually balanced.  It also lends credibility to your ability as a DM to make sure that the players know you’d never let the baddies do something that they can’t either. 

One of my favourites was this one:  

“I think if I word the Suggestion spell craftily enough it should let me tell a Goblin to go kill all his friends”

And so ask yourself - if a bunch of low-level Yuan-ti baddies did the same thing to your party, would that be fun?

Its not always about RAW, sometimes it makes sense to think critically from a different perspective. 

Alright, you want some really hard questions?

• What do you do when the players decide their PCs attack your quest giver NPC and fight to the death rather than work with that character?
• What do you do when when the players decide their PCs attack that allied NPC that had the secret because they thought the secret was that the NPC was the BBE the whole time (but it was supposed to be their Obi-wan/Yoda)?
• What do you do when the players decide their PCs let your plot hook go right past them without even so much as a pause?
• What do you do when the players decide to go shopping and you haven’t prepared a single NPC?
• What are the three most used chapters of the PHB/Basic Rules?
• What’s the best way to learn how to not make a mistake?
• What do you do if you don’t know a rule and can’t find it in under a minute?
• How do you know if you’re doing it right?
• What is rule #1?