Waching Movies in 75 to 144 hz

Well I use a Gigabyte G32QC 32" curved 1440p 165Hz monitor as both a monitor and a "TV", movies streamed in 1080p and movies from disk in 1080p play perfectly fine, in windows and exclusive full screen.
The issues you may face are to do with the monitors ability to scale displayed content up without too much distortion, a sort of gradient effect on colours, mostly affecting darker colours and black screens, but if the monitor panel is decent it shouldn't be an issue.
Go for at least an 8bit display (true 10bit if HDR is required, with over 800nits peak brightness and as many local dimming zones as possible).
Rtings.com has some very good monitor reviews and spec listings, also Techtesters has recently done a fair few monitor reviews lately delving into HDR, colour representation and brightness/ contrast.
Main thing I suggest is not to go too cheap, but also don't spend over the odds on features you probably won't even use.

The frame rate is adjusted on the view layer (the Films & TV window for example) so a 24fps film is displayed at 24fps, you can see this happening easily if you use any sort of CG animation app and run MSI Afterburner with the overlay active, if you set your animation to play at 30fps the app window reports 30fps, 60fps shows 60fps on the window, the monitor however has not changed it's refresh rate from 165Hz or changed it's resolution.
The opposite happens with a 60Hz display and, for example, a game running at 100fps, the excess frames are discarded by the GPU and not sent to the monitor and/ or sent too fast, which can cause synchronisation issues between the monitor and GPU resulting in tearing of the image and visible black frames inserted, hence technology like V-Sync, G-Sync and Free Sync, which attempt to smooth out the frame time.
Of much more importance when viewing 1080p content on a monitor is the DPI of the display, past a 24" screen size the DPI for 1080p panels can be so low that the image becomes "blurry", especially text, 32" is probably approaching the limit for 1440p and anything over that it makes sense to get a 4K or higher panel.
TVs tend to have a low DPI for the screen size, but you tend to sit a lot further back from them which somewhat counter acts the "blurry" effect.