If you see a lot of new growth it is best to divide as they will be too cramped and compete for food. You may find that some varieties are quite prolific while others never seem to put on bulblets. FYI my most prolific ones are the Pearl series and then George Slate, Morden Butterfly and Katinka. Also Dawn Star. I'm glad that the ones that are most prolific are also my favorites!
I don't think it hurts a whole lot if you don't get around to dividing the same season it looks like it needs it. If you are real busy it can wait till the next year. Sometimes those little leaves will just die away if not given room to grow. But if it is an expensive lily you might just want to separate the bulblets out and propagate them. It's always good to have a little insurance!
I don't think it is necessary to put fertilizer in the hole, at least I don't. Those bottom roots are used for stabilization. I do work in some new organic material prior to replanting. (So yes, dig the whole clump). Most of the feeding occurs via the stem roots that grow on the stem between the soil surface and the top of the bulb. That is why I use a granular slow release fertilizer around the new growth in the spring, then water it in well. B & D used to carry these slow release tabs that I liked. It was so easy to just push one or two into the soil around each bulb and where the feeding stem roots form.
One thing I forgot to mention is if I get around to it early enough (the stem and foliage is still elongating and unfurling in the spring) I apply a foliage fertilizer (Spray-n-Grow micronutrients and Bill's Perfect Fertilizer 6-11-5). I use it on everything in my gardens.
