If I run a dungeon with my box full, get a drop (capture a monster), and clear the dungeon, what happens to the drop? Are they lost, do I get a chance to make room, are they temporarily stored..?
Even though this pushes your monster correctly over the maximum capacity, you keep the monsters. However, you cannot run another dungeon (or use the egg machines) until you resolve the over-full collection by selling monsters, fusing monsters, or buying more space.
You can also overfill your collection with the PAL machine if you buy 10 monsters at once. For example, if you have 49/50 monsters, you can buy 10 at once and have 59/50. Again, you must resolve this before running another dungeon.
Note that while you can run a dungeon with a collection that is exactly full (ex: 50/50), you cannot use the egg machines in this state.
If you have a full collection and open mail that gives you a monster (ex: TAMADRA), the monster will not be added to your collection. However, it is not lost - you can open the mail again after making space available and you will receive the monster.