It looks like it is a three way switch, but I am not sure
It isn't. What I see is
That seems fairly straightforward to me
- The black wires are permanent line (live/hot/phase)
- The red wires are switched line (live/hot/phase)
- The white wires are neutral
- The green wire is a ground (earth) bond between switch housing and metal wall-box
So, I don't see a three-way arrangement (two switches controlling one light)
Whether the use of colours is compliant with local code (regulations) depends on where this is located and when it was installed.
Sorry for the horrible diagram, I must find a better tool than MS-Windows-Paint!
As bib pointed out in a comment, my diagram omits one black wire from the left switch to the bottom left (room light) cable. I conjecture that this may have been to provide power to something like a ceiling fan with a separate pull-switch built into the hub - but that's just a guess on my part.
how can I tell if the box is actually grounded?
It probably isn't, unless there is a connection to the back-box we can't see (perhaps to the outer metal sheath of an amoured cable.
You can test this with a multimeter and, with all power to the building turned off and verified off, measure resistance between metal-back box and the white wires. If the resistance is low there is a ground connection between back-box and the neutral-ground bond near the main incomer or main panel.