Applying auto colour correction settings to another image?

Yes

In GIMP the are several auto color options to try. But they just work on the image you have open, if you open another image, the auto color adjustment will be based on that image, not the one before.

Click Auto > Color > then choose one that works. If it doesn't work, hit CTRL+Z to undo, then try another one. Often these might not work well, because they are just auto adjustments, and it will depend on the image. If you want more control there are other less automatic methods such as.

Colors >

then Color Balance

or Hue-Saturation

or Levels** (the dialog also has an auto levels button)

or Curves*

*Note: If you use curves, you can save the curves adjustment as a preset, and then you can apply the same preset to each image you open, so the same color adjustment is made on every photograph you apply it to. But creating the initial curves adjustment is not automatic, it's a manual adjustment.

**Note: If you use a levels adjustment or an auto levels adjustment, you can click the "Edit these settings as curves" button, then you can save the curves as a preset, then you can apply it to multiple images.

Or if the white balance is off, you can adjust that too using Tools > GEGL Operation > Color Temperature, then adjust the sliders until you get the right colors.

Yes. In most programs, you can either save a custom setting, apply the last setting to a new image, or batch process images.