Yes you can use leftover mud if it is in good condition, for touching up rips and tears I use a 6" tapeing knife and a mud pan, to skim the ceiling I use a 10" bedding knife. For textureing, well that depends on the texture you want.
You may consider hiring a texture man to come in and do the job, he will have all the tools and know what to do with such problems that can arise when retextureing after popcorn has been removed. Popcorn texture is very vunerable to dampness and if it has not been completely removed to the drywall the moisture from the new texture or paint can loosen it and begin to peel, bubble and let go. Whenever I remove popcorn texture, I usually alway prime with an oil primer before proceeding to help stop this problem, then repairing and skimming, and of course after you re-texture, it will need to be primed again with at least a good drywall primer prior to painting.