What is the correct setting for this field? Does it need to be adjusted from one camera to another?

There is no correct or incorrect setting. Adobe created several profiles for each camera with focus on different qualities. Select profile that gives the most acceptable colors for you.

On my Nikon D750, the color profile is set to 'sRGB'. Should I leave this as such or switch to "Adobe RGB" for better color reproduction (or matching in Adobe Lightroom)?

The camera profiles in Adobe Camera Raw and Lightroom are unrelated. You only need to worry about in-camera sRGB/AdobeRGB setting if you shoot JPEG. And you only need to worry about Adobe camera profiles if you shoot raw.

I know that sRGB is the most widely used profile online, and unless one is using a wide-gamut monitor is a safe bet. But why is Lightroom then setting the color profile under camera calibration to "Adobe Standard"?

The Lightroom profiles are here to guide the conversion from raw. Every conversion from raw. The sRGB/Adobe RGB profiles are applied only after the image is converted from raw to RGB. And in LR, you only need to worry about these RGB profiles if you export your photos as JPG, TIFF or similar or if you print with manual color management.

If you are a photojournalist is changing this setting considered "manipulating" the photo?

Very unlikely. At least as long as you use the default profiles. However, you could create profiles that severely manipulate color, these could cause some problems, I would think.

If camera calibration is available only to interpret RAW images from a given camera properly, what's the correct way to set this field? Or should I leave it as such (to "Adobe Standard")?

Set it to whatever gives results that are closest to your vision. Unless you make your own profile, you will probably find that for most of your images it is either the Adobe Standard or Camera Default. But try all of them and see which one you prefer as a starting point for your further edits. Lightroom also allows creating a preset from the regular editing sliders, so you can have your individual style applied automatically on the images when you import them.