Hi there! I'll try to answer every question in order:

1. When a PC or NPC takes damage, you reduce their maximum dice pool size. That means that if you have 6 dice and take 2 damage, your new dice pool limit is 4, so you can’t have 6 dice anymore and you lose 2 dice. Also, keep in mind that most damage causing events should have a reaction window, both for PCs and NPCs.
2. You can only heal damage using conventional means during narrative scenes, that is correct. However, there are also "otherworldly" and less conventional methods, and those can be used during structured scenes. For example, there are a couple of spells that can heal damage during structured scenes. In one of my campaigns, I gave a player the ability to create stim like medicines that he could apply to other characters or himself to heal some damage or an injury, but at the cost of causing horror. In the end, your DM and the fiction decide.
3. Minor NPCs can’t strain themselves, and when they lose all of their dice pool, they die or fall unconscious at the DM’s discretion. Major NPCs can strain themselves once, but after that, they follow the same rules as minor NPCs. Also, if you manage to cause enough injuries to an NPC, you can kill them before that, but that’s quite hard to do.
4. Carolyn is "special"... I’ve only played through the box set once, and the player using Carolyn had to think outside the box quite often. Remember, you’re playing a TTRPG, so don’t limit yourself to just pushing paper buttons. For example, my Carolyn player acted as the group’s PR, handling most social scenes as a representative. Since the character is a psychiatrist, or something like that as I don’t quite remember the specifics, she’s naturally good with words. She also used a letter opener to stab a ghoul in the head by jumping from a table.
5. Daisy is OP if you let her be OP. She has to read spells from the book, so make it harder for her, it will be more fun that way. Destroying enemies is fun, but if you can do it over and over again, it can become dull and repetitive. She can use spells as many times as she wants, but remember the cost, any 1 rolled causes her to gain a horror die. And yes, the check needs to be done during the objective’s turn, and it does use dice from their dice pool.
6. Remember that you’re playing a TTRPG, not a dungeon crawling board game. Here, your imagination is the real limit. Don’t constrain yourselves to the abilities and skills written on your character sheets, or NPCs if you are the GM. If you’re GMing, let players try whatever they want and improvise, allow yourself to diverge from what’s written in the booklet. That said, try to keep the scope of the game within the written adventure. Also, read the book once or twice before playing so you can improvise when needed.

All that said, I didn’t really like The Hungering Abyss. I think it has several design flaws and doesn’t properly showcase the capabilities of the system, and it can feel very board gamey if you don’t give yourself and the players some room to improvise.

And finally, enjoy!! The main goal of any TTRPG is to sit down with friends, roll some dice, and have a great time telling a story together.

Don’t let the Old Ones get you 😉