This has happened 3 times in a row so far. First time - I was traveling the map and I see a notification on the bottom right for the kingdom. I open it up and I'm greeted with a message saying "Letter for the Ruler" failed and "The document was lost among other papers." This was a surprise to me since it was just so abrupt and I didn't know such an event was even available.

Second time - Again I was traveling the map and I see another kingdom notification. I open it up and it says "Letter from Brevoy" failed.

Third time - I'm physically in the throne room and I do one of those 2 week events to upgrade one of my kingdom stats. When the 2 weeks pass, I suddenly get a notification saying "Letter from Maegar Varn" failed.

I have no idea what's going wrong and this is wrecking my relations stat.

It's like tutorial says. You should come at the end of each month more or less.

I visit the capital more than once a month, and when I enter the throne room I never get letter notifications. These events/problems/opportunities are supposed to show up as cards on the lower left side of the kingdom management screen so you can click on them and initiate them, but for the letter related ones I never see them available there. The only time I see them is when they very suddenly me they failed.

Events can spawn every week. Ones that spawn on the 1st day can fail the same month they spawned. The rest can't. As to the letters... I've no idea. Don't even remember what it was about, so it must be trivial. Maybe you need some advisor position filled so they are like 'yo, baron, build the damn road, chop-chop!.' or some such. And IIRC merc advisors don't work for that, unless they fixed it(which I doubt).
If I;m not mistaken, only letter from Brevoy if when Shandra(or whoever you've picked) approaches you in the throne room and is bitching about Olegs taxes. And then you need to complete a card about constructing the road.

Difficult to make a mistake like this, but to exclude other things, let's go there... Have you clicked on the various tabs and sub tabs to the right of the screen when in the kingdom management screen? If the tab is constantly active on "Training" for example, you won't see the 'cards' for the problems that pop up in the "Events" tab. However, there will still be red numbers indicating new events in those categories, so it should be pretty easy to spot.

About handling them, I've found it's a decent enough tactic to send people out on 14 day missions around the 10-15th of a month. That way they get back in time for handling anything that pops up at the end of a month. And sometimes I chance it and send out people at the end of the month (20-25th), so they are back in time to repeat that cycle.