1. My exported images never look as good as the images on the Lightroom screen - what should my settings be?
Do you want to export those images for saving LR edits as non-RAW/XMP-versions, post-processing or are those images the ones you want to send to your clients?
Whenever I export images e.g. for editing outside of LR I use .psd (when I need to edit them in Photoshop) or .tiff (if editing needs to be done in a non-PSD-compatible program) as image format, sRGB color space. To ensure the dimensions of your image stay the same, leave resize to fit unchecked.
Using a lossless format like TIFF is important if you have to edit your images at some point later because formats like JPG use lossy compression. That means that everytime the image is saved, compression algorithms are run to reduce the file size. This results in data loss every time you make a change to the photo and save it (even at quality 100!).
If you don't want to edit them later, you can export them as .jpg. To achieve maximum quality exports my settings are the following:
2. I will send the images to clients via programs like WeTransfer and the clients will have the authority to print the images themselves. What should my export settings be to send my clients the images for print?
If you don't mind that your clients have the full resolution image, you can use the .jpg settings from above.
If you rather not want your clients to have the full-res image, you can activate "resize to fit" and set the desired export size. LR won't stretch your image if the defined width/height is bigger/smaller than the original image width/height ratio (example resize to fit "Width & Height": an image with a resolution of 1000x1500 px will be exported at 1000x1500 px even if the settings are w:1000 h:5000 px).
Whenever "resize to fit" is activated, I normally check "don't enlarge" to prevent LR from upsampling. In rare cases, when the original dimension of the image is a bit to low to achieve 300 PPI (or whatever density is needed) at the desired dimension, I uncheck "don't enlarge", check "Output Sharpening" (set the type to whatever you want to do with the image) and set the amount to "Standard". As far as my experience goes, LR is generally better at upsampling than the printer / other programs.