At least in the US, these are most commonly called "bowls," or, less commonly, "soup plates." They would be used for eating soup, cereal, sometimes salad, and other "messy" foods. The flat ones, or ones with a very slightly raised edge, are the ones called "plates."
This is a good question, but one that I can't answer definitively.
In my day-to-day life I've never actually heard anyone refer to bowls as "soup plates," other than in stores that sell expensive dinnerware or in posh restaurants! But "His plate is full of soup" would be strange. People just wouldn't, at least in my experience, associate soup with a plate... it could escape too easily! :-)
Not again please. In Czechia, we often (not always) use plates. There are soup plates and soup bowls. This discussion could tell you more https://www.quora.com/Is-the-common-soup-plate-been-replaced-by-bowls-and-on-the-edge-of-extinction-or-already-passed-that-point-Do-people-still-use-it
But first please read the existing discussion.
Should anyone have problems with eating soup from a soup plate, they can still imagine that someone spilled their soup from their bowl into their main course plate.
"bowl" is "miska" (and "miska polévky" is definitely a thing)
This is debatable though, because a lot of people use special deep plates for soup here. We call them "talíře" (talíře na polévku, polévkové talíře), but some non-Czechs might refer to them as "bowls" because of their deep middle part and because their culture might lack this kind of soup plates.
A soup plate is similar in size and shape to a dinner plate but has a deeper central area. A soup bowl is like a very large cup is shape, but either without a handle or sometimes with a handle on either side. Both terms are used in all parts of the English speaking world with which I am familiar.