Why does charge not flow through my body when I touch one lead of a battery? Why does charge not flow through my body when I touch one lead of a battery?
I think of it like this: a battery's leads have some potential relative to ground. When I touch one lead there is a potential difference between me and one lead. Why does current not flow through my body?
 A: 
Why does current not flow through my body?

Actually, current (charge) does flow in your example!
As you have correctly identified, a battery terminal is on certain potential with respect to the ground, and you (or your hands) are on the same potential as the ground. When you touch the battery terminal, the charge will flow to equalize the two potentials. Since capacitance of battery terminals to ground is very small, not "much charge" is needed to equalize the potentials.
In the schematic below, each battery terminal has a certain (small) capacitance to ground. Initially, their voltage to ground is
$$\varphi_1 - \varphi_0 = V_1 \qquad \text{and} \qquad \varphi_2 - \varphi_0 = V_1 + V_\text{BAT}$$
Now when you touch the positive terminal (close the switch), the current starts flowing from the capacitor through a resistor (combination of skin, body, shoes etc.) which is in $\text{M}\Omega$ range. Once the capacitor is discharged, i.e. once the potentials are equalized, the current (charge) stops flowing.
And it is not that you cannot feel this! If the capacitance was initially charged to a very high potential above the ground, a spark could occur between your hand and the battery terminal, which does hurt a little. See related discussion on sparks: Reason of sudden electric shocks?

