I think all shops propose Open Water classes in Labuan Bajo so they definitely accept beginners. You might not be taken to every spots though, depending on the conditions of the day. Just ask around. To my opinion, best place underwater in Indonesia just after Raja Ampat. There is no way Lombok can compete with Flores when it comes to diving.
In Flores, if you stay close to sea level (like Komodo, Labuan Bajo or Maumere) it is quite scorched during the dry season, not much green to be seen unless you get to higher ground. In November, I guess it's gonna depend on how early is the rainy season this year.
Lombok has the upper end when it comes to waterfall (the ones in Senaru are quite amazing) and beaches. Rent a motorbike in South Lombok and you can visit 4 or 5 amazing and mostly empty beaches in one or two days. Flores has great beaches too but you have to take boat or long ride to reach one. The access is much easier in Lombok.
On Flores side, you got more trekking options : you can climb Inerie or Egon volcano in one day and you are very likely to have the trail for yourself. Many hike available to reach traditional villages around Bajawa or to go to Wae Rebo South of Ruteng. Traditional culture is more preserved in Flores too.
If I had to pick only one I would go for Flores especially if you want a quieter place. You could also manage to get a bit of each if you limit yourself to Komodo and Bajawa in Flores for instance. You can reach Labuan Bajo from Mataram by bus and ferry easily (expect to leave Mataram in the end of the afternoon, overnight bus to Bima in East Sumbawa and a 6 or 7 hour ferry to Labuan Bajo). Or take a flight back to Bali and connect to Labuan Bajo.