The "finger-like things" are adventitious roots and thus nothing to worry about.
You could get quite a few cuttings from your plant - they only need to be 4-5cm in length, or 8cm for a tip cutting as my personal preference. Thinner stems can be a longer length. Where new growth has emerged from a cut section, as appears to be the case on both of the pictured stems, this may be severed at the point of emergence. A new pup will arise from another areole nearby.
So, for your main stem it looks like we've got about 28cm up to the first 'joint', which could easily make four or five cuttings, and then a further 37cm to the very tip to give you a tip cutting plus a further four or five mid cuttings, easily making eleven cuttings in total. The side stem looks like you could get about seven pieces out of it. The question is then whether you have enough room for another eighteen plants. (Of course you do!!!
)
Above all, use a razor sharp, freshly cleaned knife. Sterilise with 70% isopropanol between cuttings. You can lick the sap off the knife first if you're interested in how bitter it might be
Also, there's no need to discard the thinner portions of stem. It's all good, and even failure is informative.
You may want to try grafting the tips back onto the mother plant as well, as it's well worth getting your head around grafting. For example, just for fun I grafted a 1cm slice onto a stock and it's given me two nice pups.
Have fun, and good luck!
“There is a way of manipulating matter and energy so as to produce what modern scientists call 'a field of force'. The field acts on the observer and puts him in a privileged position vis-à-vis the universe. From this position he has access to the realities which are ordinarily hidden from us by time and space, matter and energy. This is what we call the Great Work."
― Jacques Bergier, quoting Fulcanelli