1) Can the friction in the system heat the water? Yes, but very very slowly. Water drawn in thru the turning pump goes thru a thermal effect (heat) when it leaves the pump & reenters the the tub.... blah blah blah
2) Is my element bad? Sounds like it from what you describe. Run another test to confirm:
Turn breakers off @ panel to tub - Set meter to Ohms lowest setting (should be around 200) - Touch a meter lead to each heater element probe (doesn't matter which meter lead color to which heater probe) - between 10 - 16 ohms & element is good. Any other reading & element is bad.
You did do your amp test with the heat switch on, correct? Amp draw for each of the 110v legs should be in the neighborhood of 15-25 amps with the heater going. One side of the 110v legs will have a slightly higher amp draw with the heater on. That will be the leg the pump is wired to. The pump is required to come on when the heater is on giving that 110v leg a slightly more amp draw.