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.