In Gloomhaven - do monsters spawn in the beginning of scenarios that have open doors/passageways to other tiles?
Yes, since rooms can be composed of multiple map tiles, you would spawn every monster in that multi-tile room.

Some overlay tiles, like Corridors, can combine multiple map tiles into a single room.


  Each scenario map is broken up into separate rooms by door overlay tiles. A room may contain more than one map tile if those tiles are connected by other overlay tiles. (Page 13, Gloomhaven Rulebook)


Thus, when following the scenario setup:


  Monsters in the starting room are placed at the beginning of a scenario. (Page 13, Gloomhaven Rulebook)


Or when revealing a new room when a door is opened:


  Immediately place overlay tiles, monsters, and money tokens as indicated in the scenario description for the revealed room. (Page 14, Gloomhaven Rulebook)


You do so no matter how many tiles that room is composed of.

It's OK to run out of a specific creature standee. You just don't spawn any more of that creature.