What is the origin of the sunken dugout in baseball?  Why do they not
have regular field-level "team areas" like in other sports?  Why is it
not like in many high school baseball fields where the players sit
behind fences, but at field level?

This does not discuss the origins of the dugout, but it may be of interest:

"Q: Why is the dugout dug out? Do coaches and players get a better
view of the action if they're seated below field level?

AG: A better view of cleat action and pants styles, but that's about
it. The real reason why the dugout is dug out is that it reduces a
potential obstruction in the views of the paying customers. An
above-ground dugout such as the kind used at the Kingdome requires fan
seating to be higher to avoid the obstruction."

Seattle Post-Intelligencer: The Answer Guy
http://seattlepi.nwsource.com/baseball/135635_aguy19.html

Well, the truth is that the manager wanted a kickback from the contractor.