"Our doctor has a lot of work."
Translation:У нашего врача много работы.
I have been thinking and I think without "есть" it sounds more natural because "есть" adds the feeling of possessing and owning, while the doctor doesn't really "own" the work, he doesn't literally "have" it, it is there and he has to do it, but he doesn't have it.
Same applies to diseases. We say "У меня простуда" (I have a cold) and never "У меня есть простуда" which sounds like you have a cold, in your pocket or so :) In this case only one is correct. In the case of doctor and work both are fine, but the other one sounds strange.
I'm a native speaker too, and for me saying "У меня есть много работы" sounds a bit weird. As if I was saying, "I have a lot of work to give to other people", not "I have a lot of work to do". But well, it depends on the region, as always, so if you think it should be accepted, we should report it :)
Because it follows the word много which works with Genitive. A good way to remember is, много = a lot of (of --> belonging --> Genitive)
E.g. много работы [f], много песка [m], много молока [n], много котов [pl].
(a lot of work, a lot of sand, a lot of milk, a lot of cats)
All these are in Genitive.