This is one of the reasons why the notion of a planned encounter is in some sense ridiculous.
The only way to control encounters is to railroad.
In my opinion you should have designed this as a single encounter with the assumption that no rest would be taken (or available) as foes organized and poured through doorways or fled before the PC's seeking reinforcements. I advice that not because it makes the encounter more balanced (though the recognition that a tribe of kobolds represents in some sense a single continuous encounter is a wise one) but because its less gamist and more believable. If you believe the vault and office exist at the same time and are spatially close to each other, why don't you believe that the inhabitants exist before the doors are open and are taking actions? This might imply that a whole tribe of kobolds is more than a single group of starting adventures need to be facing at once using the system you are using, and certainly in my case I would have waited to challenge a PC party with a tribe of kobolds, priests, and dragon pets until much later level. Instead, I'd probably lead up to this with raiding parties, individual cultists, and other splintered elements of the proximate foe as warm up to the dungeon dive.
Your PC's have done nothing more than rip your gamist veneer off your simulation and now you are wondering what is going wrong.
What is going wrong is you tried to design a tactical war game but played a role-playing game. In a tactical wargame, the scenario you have isn't possible. Encounters aren't triggered because they are scenarios. They are scenarios because there really isn't usually a chose about whether or not you play them. They follow after each other one after the other and don't occur in any larger space than what is necessary for the tactics of the encounter. Things that leave the encounter map leave it because there is no reality outside the encounter map.
You tried to create a reality outside the encounter map because you at some level believe the game is a simulation of that reality and this prevents you from having scenarios. Congratulations, you have a role playing game but unfortunately it's not clear to me that that is what you actually wanted to have.
And why did they have to open both doors?! I had everything planned for their success!
You are not in control of the story.
I want them to win, of course, so they can continue exploring the plot I've created for them.
You are not in control of the story.
If you are creating a world with real space and time so that the possibility of, "
You open two doors at the same time" is a real and present one, then the correct way to look at a situation like this is " You need them to win so they can continue creating their own story." Unfortunately, they've decided to create a story of how inexperienced and overconfident young adventures made some rash decisions that led to disaster. And now you are unhappy because you imagined for them a completely different story without getting their input. One of the cardinal sins of a dungeon master is to spend to much time imagining how something will play out in a 'cool' manner - which I'm inclined to think you did. Never try to imagine what the story will be. To the extent that is useful, you should have been imagining things like, "What can go wrong?" instead of imagining things would work like a perfectly scripted novel. It never will, least of all the perfectly scripted novel you wrote on your own.
I think your players have been rash and earned a defeat, but in the defense of your players the very fact that enemies didn't come out of the doors during the prior combat may have led them to believe that there were no nearby enemies. The very gamist assumption you made of distinct and separate combats may have lead the players to imagine their physical surroundings in a way very different than they actually were precisely because the way your physical surroundings were behaving suggested a physical reality different than the one you drew. If I make a lot of noise and nothing comes to investigate, my assumption is that there is nothing nearby to investigate. This assumption is false in this place primarily because I think your reality is incoherent.
While I also wouldn't have let burning hands spot wield hinges because this implies a level of precision to the fan of flames which isn't specified by the text, and usage this implies an amount of heat which if consistently applied would allow the spell to do all manner of fantastic things and beggars the question of why it is not more lethal to living things. I'm pretty sure that if I allowed such a creative usage of a spell that in very short order the spell would cease to have any limitations at all. It would be an all purpose fire spell that did whatever the heck the player wanted - instantly heat metal armor to scorching, heat weapons to disarm foes, fire in narrow rays through crowds to pick out individual targets, act like a temporary wall of fire, make precision cuts in wooden objects, etc. Pretty soon we'd have creative alternate uses that were always more effective than the stated use. Some people call this creative, but its trivially easy to be 'creative' when you have no limits to what is allowed. Creativity is marked by using what you have not inventing what you don't.
However, this GMing mindset I've just outline is simulationist, in that I'm assuming that whatever ruling I make here implies a physical law of the universe (namely, "Burning Hands can be used to impart enough heat to a metal object to weld it.") that implies equivalent behavior in similar situations (for example, "Burning hands can be used to impart enough heat to a metal breastplate to start cooking whatever is wearing it"). With a different GMing mindset, "Rulings are relative only to the situation and what happens is based on whether or not I think the ruling results in a fun/cool/exicting story.", you'd be inclined to rule in favor of the wielding. What this suggests to me is you are actually heavily conflicted. You really want to railroad the players to success, but you are unwilling to admit that or actually embrace your own railroading preferences. You need to decide what you really want - a railroad ride through the plot you've preferred or to actually create a world for your players to create stories in. You seem to be using the tool set of, "My job is to create a world for my players to create stories in", but your actual desired goal is, "I want my players to succeed so they can experience the story I've preferred for them."
