Basement – What I do about mouse droppings in basement ceiling where I can’t reach

I would clean out the area around the holes to give myself room to work without being showered with cellulose. Open the holes a little: make them square, and big enough to be able to get your hands through. Next, get myself a roll of R-13 fiberglass batting (currently less than $10 at Lowe's or HD), and cut patches to fill the void in the cellulose insulation: you'd probably need two or three layers, depending on what size your ceiling joists are. Stuff them through the holes and spread them out. Finally, use a couple of 1x2's to cross each of the holes, and patch them as described in this answer to this question.

I've used a shop-vac for cleaning up blown cellulose, but for disposal, not for later re-installation. It seems to me that it has a very different texture before and after: clumpy and with some structure, versus powdery, no texture.

Thermal insulation (and a vapor barrier) is applied between the interior and exterior surfaces of the home. Unless you plan on leaving your basement unheated, you wouldn't place thermal insulation between the floors. And leaving it completely unheated is a bad idea since pipes can freeze.

The only reasons to place insulation between floors is for sound or to slow the spread of fire. For fire protection, the places you'd install insulation are where fire could spread up an opening like a chase for running duct work and other utility lines.

What's left is sound protection, and that is best provided by using flooring materials like carpeting above, and then separating the drywall from the joists with something like resilient channel. Insulation in the ceiling cavity will only reduce the high frequency sounds from above. The low frequency sounds, such as loud footsteps, will travel directly through the flooring, the joists, and be rebroadcast by the drywall. Think of sound like electricity, the joists as a short, and carpeting and resilient channel as tools to prevent that short.