Hi Lori,
Size all depends, for a desktop site I'm currently using images around here: 2048 x 1365, which work well. Of course, I open them in photoshop to get the quality setting (and file size) as low as possible before I encounter any noticeable artefacts.
Screen sizes are so varied that you're probably going to encounter at least one setting where your images will pixelate. To mask pixelation, many people put a repeating pattern in front of the image (which will also prevent any right-click saving, but remember, no image online is safe from theft. They can still look in your source code and get the image URL if they really want it.) To do this, make a transparent png around 5 x 5 px and fill it with either some 1px dots or 1px wide diagonal lines (like this one I made, feel free to use it: http://exbrawladsandlasses.co.uk/images/antipixpattern.png). Then cover your images with a rectangle in muse, using your pattern as a tiled background fill. For a full-page slideshow, you'll have to make it 100% width, and make it really tall to cover large screen sizes. Set it to scroll at the same speed as the page, so that Muse does not create a scroll bar for it.
You can't automatically add watermarks with muse, and really, is there much point? They are hideous, and so easy to remove. If you really must, however, open photoshop and use the actions panel to record and automate placing your watermark on the images, then import into muse.
I hope this is helpful, tweet me if you need any more advice: @upchurchpwm
Thanks,
Nathan