Which item should I show to the guard to prove myself as an adventurer so that he will unlock the gate?

Don't bother yourself with that quest just yet. You'll run into it eventually.

Spoiler: show
When you complete the missing brother quest in the tomb you will get a key that will allow you entrance to a room with a wall tapestry thingie. Pick it up and show it to the guard. There's 2 treasure rooms that need a key. If you happen to open the wrong door then don't panic, you will find the other one as well.
Note: The proof of heroism is easy to miss if you don't know what to look for. When you find the treasure room and open it with the key then remember to look around. It's on the wall right next to the chest.

Then I seem to be at an impasse. I have completed all the quests I have come across except for "Messenger" and the main quest. The only two directions to explore further, as far as I can see, are blocked by gates, one of which is "locked permanently" while the other is where I must prove myself as a merchant or adventurer. What am I missing?

Did you click the "Spoiler: Show" button down there in my post? It'll explain it a bit better.

Okay, so, I'm stuck on the same thing, except I know where the paper is: I played through the first part in the free version, and then started over with a new party with the paid app.

When I got there, though, I used the key to open the door to the east, and I don't have a key to open the one to the west. I vaguely remember having two keys before. I must be missing something silly....

When you find the missing brother you will get a key. The other key I believe was in a big room with the pillars, the same one with the teleporter and the first skeleton dudes. From that room take a little hike to the west (to your left when facing the teleporter) and you should see a lever on the wall that opens the door next to it. I think it was that room where the key was hiding.
Either that or the big boss holds it. The big boss is the skeleton dude who appears when you get to the room with the two locked doors.

Hope that helps any.

Hmm, thanks. I loaded from a saved game and confirmed that the key I have will open either door, but is irrationally consumed in the process. (I guess it's a magic key.) I've slain the boss monster, I'm pretty sure. I'll go back and see if there was any loot I failed to pick up....

Looked around the whole dungeon; I don't think I missed anything.

At this point, I'm kinda suspecting this is a bug in the paid version. I thought maybe the idea was one would have to choose between the "adventurer" route and the "merchant" route -- the east door has much better loot. But it's not worth nearly the 100,000 the guard is asking, which seems to be intended as an impossibility. (Since I've completed all the quests, I don't think there's that kind of cash to be found.)

No, I'm foolish. I totally missed the area you describe this time around. Got it now.

Here's a very noob question: how do I unlock the doors with the keys? I have both keys, I've pulled every lever in the tomb, and I'm standing in a small room with two locked doors. How do I use the keys? I've tapped on the doors. I've opened my inventory and tapped on the keys. I've even tried dropping the keys on the ground, going to pick them up, and tapping on them while there were still on the ground. Nothing seems to work.

Thanks in advance for the help!

There are locks on the wall next to the doors - EOB style .