Would it be allowed to place the two extra monsters in an area that is completely empty?

I am going to play a rumor level with an empty starting area and have the card in play after level 2 - so would I get the extra mobs (not that it would matter much, all of the mobs still being one-shot copper creatures) in the start area?

I would say yes, provided you consider setting up the starting area to be "revealing a new area." I can certainly see that as being a valid interpretation of revealing an area, but I can't remember off the top of my head what the rules say about it. In general, yes you can place two monsters in an empty area when it is first revealed. Whether or not you choose to consider the start area as "being revealed" when the dungeon level is first set up is the only sticking point I can foresee.

Well, according to the card text, you have to place the extra mobs adjacent to existing ones.

That the start area is a new area which triggers the card is clear for me.

In fact, by the card text, not only do you need to place the new monsters adjacent to an existing monster (both next to one existing monster, IIRC), but you don't need to place them inside the newly-revealed area. I've had incidents where there was a monster right on the edge of the newly-revealed area and I wanted to place monsters from Hordes of the Things so that they spilled out into the previous area.

I'm not necessarily in favor of strict adherence to weird wording in Descent, but in this case I think it's pretty likely the intent was to make it so that you could reinforce existing monster groups, but not place monsters off in some corner near an important map element that the quest designer didn't intend to be guarded. And by extension, placing monsters in areas that don't normally contain any at all would probably be frowned upon.

On the other hand, I don't think I've seen any areas where letting the OL places the extra monsters anywhere in the area would really be a problem. The revised Horde of the Things I made for Enduring Evil lets you play a spawn card as a new area is revealed and place the spawned monsters anywhere in the new area (among other things), and that particular option hardly even gets used.

But by RAW, I don't think you get the monsters...similar to not getting all the monsters from a spawn card if there aren't legal places to put them all.