What happens inside salt bridges?
Ions leave the salt bridge and move further, without accumulating at the bridge end. For all salt bridges where there is no other reactions involved, so all simple salts in solution or molten salts.
Ions move all the way through. There is some concentration difference in salt bridge and main half cell volume. It is true for simple salt solutions and molten salts. Same ions move through half cell and salt bridge.
The speed of a wave is on the order of 100 000 km/s. The speed of ions moving is on the order of 0.001 mm/s. You can estimate the ion speed from calculating amount of ions participating in a reaction per second and dimensions of the bridge and ion density in the solution or molten salt.
Yes. But it is important to remember that there are two cancelling each other currents in the salt bridge. Thus magnetic field is close to zero.
Most important is a number of ion types. If there are just two types, such as in solutions and molten salts, then this intuition applies. If more ion types are present, and especially if ions can convert between types, things will get more complex. Another option to consider is a fuel cells where only one type of ions passes through, and electrons use outside route to complete the loop. So, cases with one negative and one positive ion types are considered here. All other collections of ions will behave very differently.
No. Ions are constantly being replenished by the bigger volume of the both half cells.

Electroneutrality strictly speaking is not conserved when the bigger volume than an atom is considered and precise measurments are available. There is a gradient of potential due to different concentration of ions. Concentration of ions is the reason for the effect you are interested in. Important parts to consider are electrodes, where ions react, thus dissapearing from what we calculate, and narrow areas, where ions' speed increase, such as in salt bridge.