How to install and use ICC profile when exporting/printing?

I feel you! I suggest you do the proofing with the specific printer pofiles in Adobe Lightroom Classic or Photoshop meanwhile, however keep in mind, that you should not export your images with the color spaces specific to the printer but rather leave it set to your working color space and embed that color space during export. That's what soft proofing is for - to ensure that your file with the bigger color space still conforms to the printer's color space without clipping color tones etc.

The missing soft proof feature in Lightroom is the reason I cannot make the transition. That's why I also just suggested the feature to be added in future releases. See here: https://community.adobe.com/t5/lightroom-ecosystem-cloud-based-ideas/make-soft-proofing-available-in-lightroom-desktop/idi-p/14730032#M10970

I don't think that is possible. It should not be necessary either. If a print service provides an icc-profile, then it means they know how to use color management and use it. That means you can send images pretty much in any color space you want, because they'll do the conversion to their profile when the image is printed.