Is it possible to use only the Region without a Location in Request Management?

Region is not an attribute, it’s a non-persistent filter that is applied to the Location LoV.

In this case, do we need to create new location every time we get a new address if we need to use this region in other places?

No exactly sure what your use case is, but for scheduling the task needs to have geolocation data. This can come from a location (i.e., the one you pick or create on the request) or directly from a map position. Of course Location is the better choice as it’s the basis for other features (e.g. PSO).

The scenario is this. Customer get requests from different locations almost every time. So they like to add map positions to task to show tasks in a map. Sometimes, geocoordinates needs to be created if they get address only. They don’t need to create new locations always since there will be more locations saved in the system then. So, using map positions is fine.
They need to use the Region to use region in selection rules, as  a coverage in contract line and filter tasks. However, this region does not work without a location. We tried to use dummy location as per the same name as the Region. Then there will be a problem in PSO since PSO look for coordinates of location not the Map position. 

Agree with your findings. As soon as you have a location, it overrules the map position (discussed in another thread here a few days ago). And you cannot specify a region standalone. Maybe worth filing an idea including your explanation of the use case.