Do I cut the stem by half like we do with tulips?
Simply, deadhead the spent bloom stalks! Make sure to avoid cutting leaves. Here are some articles that will help:
https://www.gardeningknowhow.com/ornamental/bulbs/allium/allium-post-bloom-care.htm
https://www.gardeningknowhow.com/ornamental/bulbs/allium/growing-alliums.htm
When deadheading a white or purple allium, should I only cut off the top "globe-shaped flower" and leave the long green stem OR should I cut the flower and stem all the way to the bottom of the stem?
You would cut the whole stem to the ground, but leave the foliage to nourish the bulb. They won't bloom again this year.
https://www.gardeningknowhow.com/ornamental/bulbs/allium/allium-post-bloom-care.htm