Are soffits required over kitchen cabinets?
Do I have to have soffits?

No
You can leave the mechanicals exposed. If they're hidden enough by the new, taller cabinets, nobody will know. If they're not hidden, people will know and whether you care or not is up to you.
If you're going to trim out the space above the cabinets, some sort of soffit framing will be very useful for attaching your trim and moldings to. You could, however, work out a system of attachment (possibly with pocket hole screws) that doesn't require attaching the trim to any sort of framing material, but it will probably be more work to do so than it would be to frame out with 2x2 or 2x3 and attach the trim to that.
If the soffit framing is going to be totally hidden behind wooden trim, I see no reason to drywall it. Simply build the soffit to the point where your finish trim will be at the level you want it to be then trim over it. There's no requirement for drywall between the layers of wood (or other trim material).