Can I repair hairline cracks in a solid wood exterior door with clear silicone caulk?
I don't see any reason why you couldn't use silicon caulking or why it would damage the surface.  Just know that you will not be able to stain over it again.  Some caulk is listed as paintable which might be a better bet as at least you can paint it down the road if you choose.

If you are going to want to stain the door at a later date, a wood filler product would probably work best as it will blend in well and accept the stain.

As for cleaning the door, you definitely want to remove any dirt and debris from the area you are going to fix. You will want to scrape off and remove any previous caulking that may be present too.