How can I drag-select components in a room in Altium?

You cannot start a selection box above a primitive (component, text, room), as Altium will assume you want to drag that primitive. What you can do, is enter a selection mode.

Altium's hot-key system makes this very easy, just hit s and then choose whatever mode you want (e.g. i for selecting items inside a rectangle).

But as Araho said, most of the time, rooms are just cluttering your design. I would only use them for multi-channel designs, where I have many identical circuits and want to copy the layout of one room to the rest.

Related to the question, and in response to 3 different comments above about rooms not being useful: I think rooms are great (when created by the class hierarchy of a well organized schematic design). I use them in all of my designs. Some benefits of using rooms:

1. You can turn off the "red boxes" any time you like in View Configuration Panel/View Options Tab/Object Visibility Box/ ... and click the eyeball next to "Rooms".
2. Rooms group sub-circuit components together for the initial import of footprints, which takes away the headache of searching for components that should be placed near each other (usually using a lot of cross probing)
3. Allows part placement of the group before figuring out where the best place for that sub-circuit goes in the PCB. Yes, this will probably need to be tweaked later, but getting a rough start with just the sub-circuit components is helpful
4. Allows for pre-routing of room components, before final placement in the PCB which is sometimes nice (but definitely not needed too often)
5. Copying room format (placement, routing) is really nice when it's available, typically on repeated sheets

Basically, rooms allow for better organization and better parts placement, and good parts placement will eliminate many routing challenges.

I'd strongly suggest using rooms for anything larger than the simplest of PCB layouts.

Update: A few funny things about rooms (in AD19):

• If you select and drag just a room, all of the room's components move with the room. If you select a room and some of it's components, the other components won't move when you drag.
• ... similarly, if you select inside area (drag from left to right, blue selection box) you can select components and rooms, but if you select touching rectangle (drag from right to left, green selection box) you will select the components but not the room.
• Rooms each have their own layer (independent from their components) and are only selectable if that layer is visible. "Red Box" rooms are on the top layer. Change a room, in properties, to the bottom layer and it turns blue. So, one funny behavior is that if you've moved all of the components in a given room to the bottom layer, but the room itself is still on the top layer, you won't be able to select that room unless the Top Layer is visible in the View Configuration menu (even though all of the components are on the bottom). Confusing.