I only have the Gerber files; can I open them in Altium, and somehow generate a .pcbdoc file that I can work with (route signal, pour copper planes, place components, etc.)?
Not directly in Altium, however it's bundled with Camtastic which can be used:
This is the simplified process.
File ► New ► CAM Document
File ► Import ► Gerber(s)
File ► Import ► Drill (Browse to drill file)
If you don't have a drill file you can create one by placing a via on a new PcbDoc then export it as NC drill
Tables ► Layers (assign the layer types)
Tables ► Layers Order (Confirm the logical & physical Layer order is correct)
Tables ► Layers Sets (ensure your drill span shows up here)
Tools ► Netlist ► Extract
File ► Export ► Export to PCB
Then a lot of work if you want it to be a normal set of Altium files.
It isn't as simple as importing the files and editing them directly - Altium cannot, with gerber files alone, determine information such as which pads make up a single component footprint.
Altium does have a help page detailing how you can reverse engineer from the gerbers, though.. See this page for their documentation, as it's pretty involved.