How many of the carbon–carbon bonds in the compound have staggered conformers that are all equally stable?

Answer : The 3 carbons which are bonded to have equally stable staaggered conformers.

There are 3 identical atoms of hydrogens in 2-methylhexane molecule (refer attached figure)

Therefore any degree of rotation of 60 or 180 or 300 in C-C bond will give a stable conformer of the same energy.

The image is attached for the stable staggered conformation.