Do Apple TV 4K settings override television settings?

Apple TV will not override any TV settings. No external device can. Apple TV can only set what it will output over HDMI. It is up to the TV how to take that HDMI signal as TV-input.

• Color calibration measures the output, and then tweaks the colors to match (SDR) calibration standards. It will only work for content from Apple TV — the TV itself is not affected. You will be presented with a before-and-after comparison to visually judge the difference. Then decide if you want it or not. Recommended for TVs that have not been calibrated in any other way. A calibration professional can probably do a better job, but Apple’s method is available to anyone for free, and most often an improvement.
• Match Dynamic Range will show SDR content as SDR, and HDR content as HDR, instead of forcing all content to the same dynamic range. The switch between SDR and HDR may cause a short no-image moment, as the TV adjusts, e.g. at the start and end of a movie.
• Match Frame Rate will send e.g. 24 fps content in the source as a 24 Hz stream over HDMI (instead of the default 50 Hz or 60 Hz). Your TV will do additional frame repeats or interpolation, as many TVs will refresh the pixels at e.g. 100/120 Hz for image stability. The switch from one frame rate to another may cause a short no-image moment, as the TV adjusts.
• Dolby Vision is currently the best consumer HDR standard available. Use it if it is available to you. If you like the Apple TV home screen interface and other non-video content better in SDR, then that is okay, as the Match Dynamic Range setting will still give HDR/Dolby Vision for HDR/Dolby Vision content, based on the video source.
• I would recommend Apple TV video setting HDMI Output as YCbCr, and Chroma as 4:2:0 (for HDMI 2.0 port on either side. For HDMI 2.1 on both sides any chroma setting will do).

Apple TV shows more than streaming video. The tvOS interface, the app interfaces, apps without video, screen saver, all these will be according to Video Format settings, as there is no “match content” that applies to those. Does that make sense?

why does the Apple TV even have a Format drop down menu?

Apple TV should detect the TV capabilities during setup, making the default settings close to optimal for many users. I can think of a few edge cases when you would want more options, but often there is no need to change after initial setup, if nothing else changes. It is also beneficial to see the current settings as confirmation that you get what you wanted to get, or reversely detect what is off when troubleshooting.

Color Calibration: exaggerated example but this will help me... if I set the TV to only red color washout, then I calibrate with ATV calibration tool, then I watch something from ATV (say an Apple TV+ movie, or YouTube TV, etc), will the picture be correct because I used the calibration tool, or will I see all red?

If the TV has been set to give a somewhat reddish tint, then the calibration will try to compensate for that, simplistically put as removing some red from the video signal.

if I turn [Match Dynamic Range] off and then watch a movie filmed NOT in HDR 4K/Dolby Vision, does Apple TV present in the quality of Dolby since the drop-down menu above this option is set to Dolby Vision? What will ATV upscale it to? Cuz that drop down at the top says Dolby Vision.

In the MDR:Off setting, Apple TV will use the dynamic range as set in video Format. That will trigger a “best we can do” conversion when non-matching and “upscaling”, which you may not like or perhaps you do — but not “as intended” by the content creator.

Match Frame Rate set to "on": frame rates come in as they were filmed got it. But is this different from refresh rates? I'm confused here. Is this option for fps or Hz? ... Because my TV is 120Hz. So what will happen? (I feel like I'm confused about this one). I might be mixing up fps vs. refresh??? And then if I turn this off, what will the refresh rates be if my LG is spec'd at 120Hz?

Yeah, that is confusing, as fps and Hz are used interchangeably - and they sometimes mean the same thing for progressive video. If your TV says 120 Hz, then that is the refresh rate of the screen, which will most often be higher than the maximum input it will allow for. (Marketing 101: higher numbers as perceived as better, so if input is maxed at 60 Hz and and the screen is 120 Hz, then only mention the latter.)

If you turn it off, then all content will be rendered at the Hz setting in Format, so probably 60 Hz. For example, a 24 fps movie will be upscaled to 60 Hz by Apple TV, and then further upscaled to 120 Hz in the TV. With the Match Frame Rate setting On, a 24 fps movie will be send as 24 Hz by Apple TV, and then the TV will upscale to 120 Hz in one go. The difference is rounding inaccuracy if done in two steps. Some users may be perceptible to the difference, as frames could appear to be alternating a bit longer or shorter.

why did you suggest those settings over 4.2.2? I thought 4.2.2 was better.

All streaming video is encoded as 4:2:0, so 4:2:2 will be the exact same video quality at 133% the HDMI bandwidth use. 4:4:4 will be the exact same quality at 200% the HDMI bandwidth use — no real gain. At HDMI 2.0 bandwidth limitations, that may matter for feature support, e.g. the resolution/dynamic range/frame rate combination can’t all be maxed out at the same time, to stay under the limit. With HDMI 2.1 there is plenty of bandwidth available, so it doesn’t matter then.

It depends a bit on the specifications of the TV and of the soundbar, and which component has a feature bottleneck, if any, but the way you have it sounds pretty good. Either way could be good.

Apple TV →[HDMI]→ TV →[HDMI-ARC/eARC]→ soundbar

good if the TV has eARC, else audio might be limited

Apple TV →[HDMI]→ soundbar →[HDMI-ARC/eARC]→ TV

needs good [HDMI 2.0/2.1] video passthrough on the soundbar, with support for DV

With Match on, we're talking HDR/SDR, and frame rate (24fps). Resolution is not a factor here. Yes/No?

Correct.

ATV will […] give me the 4K resolution, but the HDR part of it will be whatever the source provides (unless I turn Match off). Yes/No?

Correct.

Why does the Dolby Vision option in the ATV Format menu say Dolby@60? I thought the "@60" part would refer to the resolution and not the Dolby/HDR part.

@60 signifies 60 Hz (60 fps). Not the resolution (4K). The setting tries to combine several video properties in one setting (resolution + dynamic range + frame rate).

60 Hz is appropriate for NTSC countries, and 50 Hz is appropriate for PAL/SECAM countries, as that matches the broadcast frame rate and AC outlet frequency and thus usually the native frequency of the most common TVs there, although many modern TVs can do both now. The differences are historical for analog TV reasons.

No, resolution is send over HDMI as set in the Format. If you set 480p, then you’ll get 480p.

Only frame rate and dynamic range can be automatically matched.