Szczepan Bialek wrote:
"balun definition
electronics
A transformer connected between a balanced source or load and an unbalanced
source or load. A balanced line has two conductors, with equal currents in
opposite directions. The unbalanced line has just one conductor; the current
in it returns via a common ground or earth path. "

True

The braid of the coax and the radial are connected. They work as the ground
(or counterpoise).

No. There is no radial. There are two poles in a dipole and neither
of them is connected to the braid. It may be that one of them is
connected to the coax center conductor but this depends on the construction
of the balun (and the transformation ratio).

Is the voltage the same as in the live conductor (at the feed points)?

Normally not, in a 1:1 balun.