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I started painting and playing with Leviathan released. I've only done a small handful of games and haven't had a lot of time with Vehicles/Monsters.

Played a game over the weekend where I was getting confused on the rule for Big Guns Never Tire. I kept re-reading over in the core rules and I know that

- Monster/Vehicles are eligible to shoot even while they are currently engaged with enemy units, with -1 to Hit
- Monster/Vehicles currently in melee with an enemy unit can either target the unit they're engaged with or another enemy unit.

The scenario that got the gears stuck in my head was:

My opponent had a Carnifex with a heavy venom cannon. She wanted to shoot my Heavy Intercessors who were currently locked in melee with a Psychophage across the map. I was getting confused on whether or not Big Guns Never Tire applied to that specific scenario -- was the Carnifex eligible to shoot the Heavy Intercessors or not? The Carnifex itself was not currently locked in melee with any enemy units. As I type this out and re-read it I feel like the Carnifex should not have been able to shoot the Intercessors, but the Psychophage still could

Correct.

You can shoot out of combat or into a combat you’re part of, as a monster or vehicle.
But you can’t shoot into another model’s combat.

Yeah, makes sense now after typing it out. Sometimes I just need to talk it out, I guess.

It would suck if a whole army of Monsters could just shoot at whoever they wanted at all times

You are likely confusing the part of the rule that allows you to shoot at an enemy vehicle or monster that is in engagement range of a friendly unit, with a penalty.

Let me propose a scenario and make sure I fully understand the rule.

My Leman Russ is engaged in melee with a squad of Hormagaunts that also charged my Ogryns. Up the battlefield my squad of Rough Riders is engaged in melee with a Carnifex and a squad of Genestealers. There is also a Warrior squad left unengaged. When my Russ shoots:

1. I can shoot my Heavy Bolters into the Hormagaunts even though they're in melee with my Ogryns, because my Russ is itself engaged with the Hormagaunts.
2. I cannot shoot my Heavy Bolters into the Genestealers because they're in melee with my cavalry unit (but not also engaged with my Russ).
3. I can still shoot my Lascannon into the Carnifex because it is a monster, making it also subject to BGNT.
4. I cannot shoot my Battlecannon into any unit besides the Warriors, because blast weapons can never be shot into melee even with BGNT.
5. Every shot above takes a -1 penalty to hit, because the Russ is in melee.
6. The Warriors can later shoot their Venom Cannon at the Russ, but will also suffer the -1 hit penalty because their target is in melee.

Do I have all of those correct?

You are correct. For 5 however remember that pistol weapons are not subject to the -1 to hit even if the Russ has none.