In professional tournaments/championships and across sports, can different games be started at the same time?
When or why would the organisers do that?
Can they? Yes, unless a particular regulation says otherwise. Will they? It depends on the tournament's needs.
A tournament may start matches at identical times for competitive purposes (teams cannot easily plan to attain a specific result based on corresponding game results elsewhere in the competition) or for scheduling (if a large number of matches must be completed in a short time, having a common timetable across several venues is helpful).
A tournament may start matches at staggered times for management purposes (grounds staff or officials may be necessary to begin, but not required after that; however there are too many matches to only play them consecutively).
A tournament may also start matches consecutively, where one finished before the next. This is primarily due to broadcast requirements (the audience does not have to pick and watch only one or flicker between two games) but in some sports may also be due to venue availability (for example, if there are few games each day they can be spaced reasonably on a single field).
The same tournament may apply any or all of these, as well, depending on the stage they are at. For example, the FIFA World Cup plays the group matches at staggered or consecutive times, although the final group matches are played at identical times, while the knockout matches are played strictly consecutively (even on different days entirely).