Hi! I have travelled in most countries you plan to go to (except South America), worked in some of them. Last year I travelled the opposite way around the world, visiting several of the places you want to go to.
For a RTW flight, shop around, as there are many different options, but probably none that ideally fits your travel plans. I travelled using my accumulated miles with "Star Alliance". The alliance e.g. doesn't have a direct flight from Nairobi to Bangkok, but there is one from Johannesburg to Singapore. I got a maximum of 10 flight segments on my RTW flight, and a maximum of 7 stops-overs. Depending on with whom you book, you have / don't have the option of flying to one place, then travel overland and continuing from another airport. If you do, overland trips are normally counted as a flight segment!
Kenya, Tanzania - I would not necessarily travel in both countries if you want to go on animal safari. Kenya is better for distances (they are often huge in Tanzania). Neither country is cheap if you visit parks! You have to choose the right time to go on animal safaris. In Kenya, July - September (dry season, migration) is usually good.
Unlike e.g. in South-East Asia, not much mid-range accommodation is available in Kenyan towns. Normally you either pay a lot for the night, or you end up at a probably dirty place with security problems, especially for muzungus (white people). Zanzibar is an exception, in my view a great place to visit (there is a high speed boat linking the island to Dar-es-Salaam. Stay somewhere in Zanzibar's old stone town, lose your way in the very narrow alleys, visit the early morning market, pick up well grilled, spicy street food and maybe rent a motorbike to see the island, its spice plantations, rare monkeys in the forest and the beaches.
If you don't go to Madagascar, you don't need 6 weeks in Africa. A one-week safari is usually enough (and tiring enough). One week in Zanzibar should be fine, too. If you also visit Dar-es-Salaam and Mombasa, 4 weeks overall should suffice.
Make sure you take good anti-Malaria tablets (better not "Lariam", as it often has stron side-effects). The East-African coast and Zanzibar are considered high-risk.
Thailand is an easy place to travel, with a lot of culture to see (and a lot of offers of prostitution). Avoid places like Pattaya and Phuket; they are tourist traps. For beaches you can also go to islands like Ko Samui, Ko Samet or more remote places. If you like mountainous areas you could visit the hill tribe area in the very North. Rent a car in Chiang Mai and do the so-called Mae Hong Son loop, (including the spectacular, and not undangerous dirt-track road leading West from Doi Inthanon National Park). Leave early on the day you drive it, so you have daylight all along.
Laos is cheaper than Thailand, but also a bit more difficult to travel. The next (mostly good humoured) rip-off is usually just around the next corner. If you are told there are no buses to where you want to go, that you'll have to rent a van instead, you better verify. Luang Prabang, the former Lao capital in the middle of the jungle, is a great place to visit (World Heritage), reachable by bus from Vientiane and by boat (2 days on the Mekong) from Thailand (I would not use speedboats, there are many instances of bad accidents), and of course by plane.
I expect you'd like to see Angkor Wat in Cambodia. You can get a Cambodian Visa in Bangkok. Most package tours are pricy. Overland travel takes time on the Cambodian side (and may be impossible during the rainy season). When Thais and Khmer have another go of shooting at each other, the border is usually closed for a while.
If you can, avoid crossing the land border between Cambodia and Vietnam.
In SE Asia you don't need to preventatively take anti-Malaria tablets. But if you go to remote places, you should have emergency treatment for Asian malaria with you. These days, the main risk in SE Asia is Dengue fever; never take Asprins if you suspect you caught it, use Paracetamol against the fever and a.s.a.p. go to the best hospital there is).
If you want to see parts of Australia other than the East coast, the outback is sure the place to go. You'll need a car there, as buses just cruise on the few main roads from town to town, or rather from tiny village to tiny village (except the one going from Darwin to Kakadu Park, a place I definitely recommend). If you decide to travel from Darwin to Perth (Karijini Park inland is fantastic, Kalbarri I'd recommend too), 8 weeks should be just enough. Of course you could also go to Alice Springs to see Ayers Rock, but it's a long drive, with less to see than in the West. If you have a couple of days in Perth, see Freemantle and if you like wines, visit Swan Valley for some tasting.
Car rental is expensive in Australia. Buying and selling a car will take time and won't be cheap either.
In the outback you should be prepared to sleep outdoors, so either have a car in which you can bunk or take a tent (close it at night, to keep poisonous visitors out). And of course don't ever lose your way; it would likely be the last time.
Avoid the rainy season; I was stuck for three days at a remote camp-site once, as the road back was under high water.
I am going to New Zealand again this December. I wish I had more than 4 weeks. Not enough time to do hiking.
If you just want to relax in the US before returning to the USA, I think Las Vegas would be more suitable than hectic New York (and cheaper, if you manage not to gamble).
Good luck and have fun!