I'm a new DM running Lost Mines of Phandelver with a group of new players. We are all comfortable with not sticking 100% to the rules as we try and feel out what it means to play D&D 5e and what's fun. This website has been a great resource for answering my questions as they crop up. Now I have one that I haven't been able to wrap my head around from searching the internet...
There are two scenarios in a dungeon crawl, both related to initiative, that I would like some advice with.
1) After a battle in a room (e.g. Cragmaw Castle in LMoP), are all initiative rolls and the turn order wiped clean? Or is it typical for the same battle turn order to be used for the subsequent non-combat activities (moving around, inspecting things, etc.)?
In an example where a monster is hidden in the room, and someone discovers it with a perception check. Would I then roll initiative for that hidden monster and insert them into the turn order? Or does everyone re-roll initiative for this new combat?
I think the root of my question here is how do I respect the fact that some action has occurred to introduce an enemy into the scene (e.g. a character pulling back a curtain and suddenly is face to face with a waiting hobgoblin) while also respecting the rules of initiative?
2) I have a player who roleplays a rather impetuous and foolhardy fellow. Against better judgement, he is often barging in to rooms without really doing any checks for what may be on the other side of the door. What sort of questions should I be asking him about this action? He wants to be able to say "I yank the door open and charge in, attacking the first enemy I see.", but this feels wrong to me - to give the character carte blanche to suddenly attack.
Thanks, and please let me know if I can improve how I ask these questions.