3. If it is really two bridges, then I wonder, if these must be built from the two different sides?
Yes, each bridge must be built on its own city tile, as that's where the moat is.
4. Hmm, what about walls in between the two adjacent moats, after the bridge is built?
They stay in place - walls always are placed on a hexside.
5. And what about power lines, do they need a bridge, to cross a city moat?
Yes. Just like a river, a city moat divides the hex in two until a bridge is built.
6. Do you need a bridge to destroy a wall from within a city?
Yes (see above).