1 : you can easily dive 2 or 3 days between Amed and Tulamben. You can find some walls and some black sand slope.
2: Pemutearn is on the northwest, but Candidasa is south east. Pemuteran is far from the rest of Bali, 3 to 4 hours drive with car. I would not lose hours in car ( Pemuteran is not better than North Sulawesie) .
3: Sanur or Amed/Tulamben have OK beach ( Sanur has white sand while east Bali has black sand) , you can find good dive centers at both locations.
4: Bangka is nice, but there is plenty of nice resorts on main island cost like Kima Bajo or Cocotinos. I would do a mix of both coral and black sand ( see answer to question 7).
5: Any agency in Indonesia can do it, just pick out the one that looks serious to you. Garuda Airline can be bought outside Indonesia .
6 : muck is easy diving, mostly black volcanic sand going slowly, looking for critters, not deep dive. Critters are rare species that you will generally not find anywhere else, the list can be long, have a look at some photo galleries on the Lembeh resort web page.
7: To be able to answer you the best, we would need to know your arrival and departure time in Bali, so it might or might not connect to Manado or from Manado.
Another option is to flight with Singapore airline and arrive ( from Denver) in Denpasar and depart from Manado .
I would do Bali first and the AOW in Sanur or/and Amed few days and then go to north Sulawesi for 5/6 days coral and 2/3 days black sand.