Dragon’s landing parachute is stowed roughly in the nose cone hinge area, and this leads me to believe the nose cone has to open for the chute to deploy.
That belief is incorrect. Other than both being installed near the top of the vehicle, the parachutes and the nose cone have nothing to do with each other.
In normal operations, I believe the nose cone is ejected prior to atmospheric reentry, but on launch escapes the nose cone would be locked shut.
The opposite is true: the nose cone is open until after the deorbit burn, since the deorbit burn uses the Draco thrusters mounted in the nose cone.
The nose cone hatch is closed after the deorbit burn. In case it doesn't close, it can be jettisoned, and the Dragon can reenter without it. But that is a contingency option, since it makes the specific harder, if not impossible, to reuse.
So I’m wondering: does the lose cone have to be ejected for the chute to deploy, is it enough for it to be opened (but still attached), or can the chute deploy even with the nose cone closed?
In normal operations, the nose cone hatch will always be closed when the chutes deploy, unless it was jettisoned after the deorbit burn.
The big thing I’m trying to figure out: will the nose cone be ejected during launch escape of one is triggered? I’m trying simulate those and if I’d like to get this right—I don’t want to send my nose cone falling if this isn’t something that would normally happen :D
The nose cone stays closed for parachute deployment. IIRC, you can see that clearly both in the Pad Abort Test and the In-Flight Abort Test.
You can also clearly see it on the videos of the return of CRS-21, Demo-1, and Demo-2, as well as the parachute tests.
In case you can't clearly see the deployment sequence, it is also easy to verify in the post-recovery pictures, you can also see clearly that the nose cone is still attached: In-Flight Abort Test, Demo-1, and Demo-2.