Hello Julie- So many places so little time
This may be one of the toughest questions anyone has ever asked on Travel Blog.
With 10 days your time frame it sets limitations but you will have a great time. I'm not going to address the financial limitations in my answer.
If I were following your guidelines I would narrow it to one country per continent and maybe stay in one city or two at the max
It sounds as if you have a variety of interest and I would select a variation between cities and World Heritage Sights.
Australia - We spent two months there and didn't see half of what we wanted. Sydney is a great city. Tasmania was one of our favorite places. We loved the west Coast. I would go to Uluru and Tasmania for sure.
Asia - Again I would narrow it down. Example: 9 days in Cambodia you could see Angkor Wat and the countryside
You could go to Borneo and see the monkey's
Thailand if full of great architecture and islands for diving and sailing.
Africa - So far we've only been to Morocco. It was wonderful
Antarctica- I worked in Antarctica for 5 months and enjoyed it. You may want to check out that possibility.
Europe - I'd pick one country and two cities. Paris is amazing. You could do 5 days in Paris and 4 days in the Normandy area.
We loved Iceland and Greenland. We love Greece, Athens is amazing and the islands are breathtaking.
Belgium has some amazing towns.
North America - My favorite U.S. cities are New York, San Francisco and Washington, D.C.
The National Parks are amazing. Utah Canyons are unique, Yellowstone and Grand Tetons are beautiful and Colorado's Rockies are a joy.
South America - We loved every bit of Peru.
New Zealand is full of hiking!
I will think about this question and add more later.