For the ceiling joints, the mud will probably not adhere well to the painted surface, now, so the paint and some of the old mud should be removed to expose the joint compound underneath the paint. Might as well remove it all and re-tape as RXPLXQRTSMNLOP... ( :-) )suggested. Wet the joints with water, let them soak, scrape the old mud out, and re-tape.
Do not bother trying to "cover" sand texture with a "smooth" texture; remove it instead, if possible. (Primer paint will not cover it up until you use twenty layers or so.) Sand texture might be from actual sand mixed into the paint or it might be just dried flakes of joint compound/gypsum. You can use sandpaper to sand it out, but that can get dusty. Depending on the paint used, it often can also be removed by soaking the wall with water, letting it absorb for at least fifteen minutes (and re-wet it every few minutes to keep it wet as it soaks), and then scraping gently with a drywall knife. (A drywall knife has a naturally smooth edge that avoids cutting). Make sure you use a small angle with the drywall knife so that it is almost parallel to the surface. Otherwise, the drywall knife will cut into the paper on the drywall and rip it.
If that does not work, or if it seems like the wall nees more repair, you might just want to rip it down and re-drywall the whole thing. Many people (including me) tend to miscalculate how much time and effort fixing a bad wall/ceiling will take. I added several weeks onto my renovation by trying to repair and re-use the original ceiling drywall when I should have just ripped it down and bought new stuff (which I ended up doing). It is often cheaper and easier (and better) to just replace it.
As for your painting question, spraying is a much easier method of painting, especially for painting ceilings. I have used a paint stick, a plain roller, and a sprayer, and the sprayer was by far the easiest. Holding a roller up above your head, especially one filled with paint like those paint sticks, is tough work over time.
[This message has been edited by Lawrence (edited March 05, 2001).]