Possible to combine Kilimanjaro, Ngorongoro with a gorilla safari?

Not only is Burundi not safe at the moment but also they don't have gorillas - when they promote gorillas it's actually into neighbouring Rwanda or DRC. DRC is your 3rd option for gorillas but to get there you'd have to go via Rwanda anyway. Rwanda has the most expensive permit but is the easiest place to access the gorillas given your short time.
So just fly into Kigali (either at the start or end of your trip) from here it's about 3hrs to Ruhengeri which is the base to go see the gorillas. You'll have to get your permit in advance. For Uganda you'd have to fly to Entebbe from where it's a very long drive to Bwindi (all day).
Probably the most logical would be to start with Rwanda then fly Kigali to Arusha for Kili/Safari then fly to Zanzibar for some beach time at the end.
Actually, thinking about DRC, you could also climb the volcano there, and see chimps (http://visitvirunga.org/). You do need to keep an eye on the situation in DRC however as it comes and goes in terms of safety - rangers were killed recently by rebels in the park here.

Thanks. I have done some more reading and Rwanda indeed seems like the best option. How much time do you think we would need for gorillas in Rwanda, Kilimanjaro and Ngorongoro? One option is to skip Ngorongoro as we will probably do safari in Etosha (Namibia) during September.

Rwanda you will need about 3 days and Kilimanjaro climb will take 6-7 days so plan with 10-12 days will be good. Your flight from Kigali to Kilimanjaro might be a late one so you might need a night in Arusha. Ngorongoro crater is very special if you can please go there you will not be disappointed.

As above. In Rwanda
day 1. Arrive and transfer to Ruhengeri
day 2. Gorillas
day 3. transfer to Kigali and fly out
At some point, either day 1 or 3 you should visit the Genocide Memorial Museum in Kigali, really powerful - you can plan it around your flight.
You could also spend another day and go on the Golden monkeys trek, or another trek.

Kili, depend on route but about a week.

Ngorongoro/Serengeti, this could vary but probably 4 days

day 1. Arusha to Serngeti
day 2. Serengeti, transfer to Ngorongoro
day 3. Ngorongoro (you could then transfer back to Arusha or maybe throw in a stop at Lake Manyara)
day 4. to Arusha, fly to Zanzibar.

So 2 weeks for Gorillas, Kili and safari, then a few days in Zanzibar (spend a night in Stonetown as well as going to the beach).

Worth noting the closest airport to Kilimanjaro is, unsurprisingly Kilimanjaro Airport which is closer to Moshi, not sure what flights are like between there and Kigali though.

Anybody who has been to both Etosha and Nogorongoro, Serengeti, etc?

Personally having been to both, I'd choose Serengeti and Ngorongoro by a big margin.

I've been to Ngorongoro and Serengeti a couple times, I've been to Etosha 3 times. Both great destinations, both different. Etosha's big thing is the wateringholes, flood-lit at night, sounds horrible but actually really good, it's like being at the theatre and just watching animals coming and going.
Ngorongoro is just amazing because of the concentration of wildlife. Sernegeti is also fantastic.
Even though you'll have gone to Etosha I would still definitely go on Safari in Tanzania. For what it's worth I saw leopards both times I was on safari in that part of Tanzania - doesn't mean you'll see one, they're one of the harder of the big animals to see but it's a good place to see them.
At the end of the day, it also comes down to your own interests. I'd be more than happy to go on safari in just about any park any time. Even going back to the same place. Every game drive is different. One of the most amazing 24hrs I've had on safari was in the Serengeti, we saw lions in trees, saw a leopard, saw cheetahs, AND saw caracal and serval. That's just about every cat apart from the African wild cat (which I've never seen in the wild but is the hardest of the bunch). The people I was with were on their first safari and didn't quite believe when I told them this was not usual and they were incredibly lucky.