Ok, I installed a new wiring harness. I bought it from Walck's.
Questions #1;
Is it normal that I have current even when the key is not at the ignition position or not there at all?
I'm able to turn on the light even if the key is removed...

I'm pretty sure that I wired it ok up to the key switch.
The only screw identified on the key switch is marked with a T. There is three screw. I assumed this one was the battery screw.
The key switch has a left, center and right position. I wired it to have power when the key is turned to the right. I don't understand.

I feel stupid.

Question #2;
The battery negative pole was connected on the fender. Is it the right place?
Question #3;
I got a Sparton SCS-3R signal turn switch. I installed it. The pilot light on it does not light up.
I can see (not able to measure it) voltage going to it but I think that the new flasher is too fast.
The bulb is good. It's a 53. Even if it's for a 12 volts, it should light up, right?
I will try to find a 51 and give it a try.

Question #4;
All my light worked even when using the turn signal except for the brake light.
The tail lights are on and I can see it flashing left and right but I have nothing if I hit the brake.
Is there a way to test the brake switch? I guess I can bypass it and connect the wire together.
Is there any ajustment on it?