Is flexible metal conduit rated to be grounded?
What fitting is best to connect to metal box?
Do I use bushings?

So if I branch off one box to a switch box, I have to run a separate ground as well or does the mechanical ground between the boxes and connector suffice?

Flexible metallic conduit is not rated to be the mechanical circuit ground

Flexible conduit will automatically be grounded when it is connected to the grounded metal box.

So which is it? Do I need to run a ground wire between the two boxes to achieve my mechanical ground?

Flexible metallic conduit is not rated to be the mechanical circuit ground and must have a ground wire pulled through it to the device box.

What material should I use here flexible metal or AC?

I thought that you could go 6' using FMC as the grounding conductor?

Where flexibility is not required after installation of the FMC, a grounding conductor is not required and...

(5) Listed flexible metal conduit meeting all the following
conditions:
a. The conduit is terminated in listed fittings.
b. The circuit conductors contained in the conduit are
protected by overcurrent devices rated at 20 amperes
or less.
c. The combined length of flexible metal conduit and
flexible metallic tubing and liquidtight flexible metal
conduit in the same ground-fault current path does
not exceed 1.8 m (6 ft).
d. If used to connect equipment where flexibility is necessary
to minimize the transmission of vibration from
equipment or to provide flexibility for equipment that
requires movement after installation, an equipment
grounding conductor shall be installed.