Fender Mounted Turn Signals

What would happen if lights were installed in the dash? Would there be too much load on the flasher? Would the lights flash extremely fast?

They should work fine. Even if they didn't (they draw very little extra load) you can buy a "heavy duty" flasher. Back in the day, this used to be a "552"

My '70 RR had the usual front lamps, TWO rear lamps per side, both dash indicators and hood mounts, and I sometimes pulled a trailer, adding 1 more lamp per side.