Yes. The difference is that the lighted version does not require stopping (because it's only lit when there is no cross-traffic anyway), whereas the static sign does require a stop before turning.

This. I was actually unaware of this distinction before studying for the driver's license and always assumed the sign was just a stylised version of the light to save energy.

Also worth of note: when traffic lights are active they always take precedence over static signs which in turn take precedence over road markings (which themselves take precedence over road layout). The green arrow sign is special in this regard because the rules for traffic lights explicitly take it into account -- otherwise the red light would always trump the sign because of precedence rules.

This is the normal sign, yes. But in some towns (esp. Dresden, but now everywhere) we have simplier ones, a green arrow without light. You are allowed to turn right on red then, if traffic allows. Similar to the US.

https://ais.badische-zeitung.de/piece/00/bb/82/9b/12288667.j...

Not in this form, no. The green arrow light would be required to be next to the other green light in this case (in Germany at least). But those do exist, yes.