I believe this is a Euphorbia lactea. I can cut the long stem near the base of the plant without killing it right? Also, I believe the cutting(s) can start new plants, is that correct? Rooting powder necessary? Any advice appreciated.
I have propagated Euphorbia via cuttings many times. Sometimes you can even get root formation before taking the cutting by making a partial cut and applying rooting hormone.
My mother has one of those. She’s given away rooted cuttings to most of her friends by now. You won’t kill the original, and you’ll likely end up doubling your initial population. 
I have one of these. It is best to prune it where it branches off, but you can cut the main stem and it will survive.
I don’t know if a cutting of the main “trunk” will root, but certainly cut branches will root easily. When I lived in an apartment, my Euphorbia would get too big and every few years I would cut off a branch, get rid of the main plant, and then plant the branch in its place. I never bothered with rooting hormone.
The best method is to leave the cuttings on the windowsill for a few weeks to let the wound heal and form a callous. After that you can readily plant them and water sparingly until roots form.
Please be very careful with the sap. Wash your hands and tools carefully. Euphorbia saps very in toxicity throughout the genus, but I tool someone to the hospital once due to some pencil tree sap getting in their eye.
Thanks, everyone! I will be careful.
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Yes, good advice here.
I’m not sure you need to wait several weeks before planting the cutting - you do need at least a few days to let the cut end dry out and callus over some, in order to avoid the cutting rotting when you plant it. I’d also avoid keeping the soil in the cutting’s pot too damp - moist but not dripping wet until the cutting is established and showing new growth, then slowly decrease the amount of water you give it.
When trimming, just make sure there are buds/branches below the point where you cut. The picture makes it look like you can cut off most of that long stem since there appears to be more compact growth elsewhere.
I’ve propagated some large-flowered crown of thorns Euphorbias this way, and currently have several pots of cuttings going.