The problem with the right turn signal is not the multi function switch, and it is not the ignition switch. This kind of problem happens when one of the turn signal bulbs is either burned out, or improperly installed, or is the wrong type bulb. It can also be caused by a poor connection in the ground wire from one of the light housings. The flasher is sensitive to the amount of power drawn by the turn signal bulbs. If one of the bulbs is not working; the flasher will speed up when you turn the signal switch to that side.
The 12.5 volt reading on the battery (because it was measured with no load on the battery) is meaningless. The proper way to test a battery is by reading the voltage while the key is turned to the start position. Another way to test the battery is to turn on the headlights, and have someone watch them while you try to start the engine. If the lights dim way down, or go out entirely; then there is either a loose or corroded connection at the battery cables, or the battery is too weak to reliably start the engine.
But if the lights stay bright, while the starter does not turn over; then power is not getting to the starter. This can come from a defective clutch inhibitor switch (on a manual transmission vehicle) or a defective neutral safety switch (on an automatic transmission vehicle). It also can come from a bad starter solenoid. And it can come from a defective start contact in the ignition switch. You can test this by connecting a remote starter switch between the battery cable at the starter, and the small terminal on the starter solenoid. Turn the key in the ignition to the position where the dashboard warning lights come on, and leave it there. Then tap the remote starter switch; the starter should crank. If it cranks; repeat this a series of times. If it cranks every time you hit the remote starter; then the solenoid is good. If you short the two wires together that go to the clutch switch, or to the neutral safety switch; you can see if this fixes the starting problem. And if the battery, those switches and the solenoid all test good; then the ignition switch is probably defective.