Player knowledge & rolls

Here is a set of related questions I have thought about several times.

1.Can/should players roll ST for npc targets for their characters actions? (Lighten DM load)

2. Should DM post npc stats (ex AC & ST at being of combat) so players know when they hit and if they are effected (fail save)

3. Should the DM show all rolls?

I know some consider that meta gaming but I feel it is just expeditious. As a DM I often post NPC ac after the first round or first hit but have not done do with ST (may as that next time DM).

I don't believe that the DM should show all rolls. In particular, if something is going on that the PCs know nothing about: Did the technician on the bad guy's ship repair that damaged item yet? Did the creature make its stealth check while it crept up on the PCs.

For combat, I do tend to show what the bad guys are rolling so that if/when the PCs get slammed, they can see for themselves that, yes, they got slammed.

I think it depends a lot on
1. The players
2. The stakes

I know players who would take that actions/stats info to make life easier for the GM, but there are other who try their best to meta-game or min/max their actions based on whatever they get to know. So that would be my first consideration.

If the group is going into some random encounter along their journey, I wouldn't bother with keeping things secret. But for the main fight against the big bad, I can see the fun in keeping some things secret and possible tie it to some kind of information roll. If you take a second to size up the opponent, maybe you can figure out its defense. Both for the GM as for the players, having some more mystery can add to the vibe.

In D&D for example, it's quite easy to get to know a creature's AC. Maybe it'll take you 5 hit attempts tops? Why not speed things up when it's convenient?

I tend to show all my rolls as a GM. Well combat rolls at least. I also quite like letting the players roll the damage they take. Is that mean? :D

If I ran a system with opposed rolls to resolve an action, I wouldn't mind them rolling both at the same time to speed up play.

I often post the most significant stats to ease play for the PCs. Though when they first encounter an enemy (especially if it is something weird they haven't encountered before) I usually play with hidden info on the first round or two. But once they've hit once the mystery of whether the enemy has 17 or 18 AC isn't worth the hassle of keeping it from the players.

I usually show most rolls. Sometimes I'll roll some rolls secret if the players are oblivious to the danger they are in. But once it can affect them directly I just roll openly. At some point I'll be sloppy and forget to change visibility anyway so might as well remove the need to do so :P

Most systems have something that says 'you should only roll if the situation is dangerous or if the outcome is uncertain. There's no point in hiding those rolls. BUT. I only tell them after they're committed. They don't get to know before they're decided, but they do get to know before they roll.

Plus, most of the time, I find hiding the rolls has the opposite effect intended. Want to make a monster scary? If you hide a monster's attack rolls and only say hit or miss, you might as well flip a coin. It's the same effect for the players. Instead, show them exactly what bonus the monster gets get. Yes, it can absolutely hit the wizard on a 6. Keep that paladin frontline. What's it's AC? If you hide it, it is the players that are robbed of info, but it's the characters that suffer from it. Its harder to roleplay genuine reactions to a monster when you don't know how much of a challenge it is for your character, personally. You need the defenses to compare to your attack to do that.

No. Maybe in real-time play. But in play by post, you don't really get any benefit from this: it's not going to make the game go any faster, as the GM still has to narrate the results.
No. It removes tension/immersion for those who pay attention and read such things (the minority), and for the majority of players, they're not going to read/remember/go back and reference things posted days/weeks ago in PbP.
No, of course not. @ForeverDED answered this well:
Show the rolls for things the PCs can perceive, not the things they can't. In an encounter, that means you show the rolls for the participants that the PCs know are there. You don't show the rolls of undetected combatants that are sneaking away/creeping up on the PCs, nor the NPC who is remote-controlling the programmed diplomat negotiating the trade deal with the PCs (though you would show the diplomat's rolls). Out of an encounter, you don't show things that the PCs don't know, like the outcomes of battles in the neighboring kingdom, even though the effects will be felt by the PCs later (they'll hear the news and see the refugees in time).

Basically, it's important in PbP to show the players rolls for things that the PCs can know, and to not overload them with a bunch of extra information that might overwhelm them, confuse them as to what is actionable/relevant, or be noise for the few players that actually go back and review what has been posted in the past.

Of course this varies by system: some RPGs expect more or less information for the players than their PCs have, and some games like Numenera already have the players making all the rolls.

Gamers Plane allows us GMs the ability to hide the reason for rolls and/or the results of rolls, both of which are useful in particular kinds of games: hiding the reason for rolls is good for Paranoia (just to alert the players that something is happening that they don't know), and hiding the results would be good for games like Birthright (where players know what sorts of things are happening every kingdom turn, but shouldn't know the results).