The formula to find the area of a circle is pi times the radius squared. I use this to change recipes to fit different pan sizes all the time.
The radius of a 10 inch round pan is 5 inches. Square this, and get 25 square inches. Multiply this by pi, 3.14 and get 78.5 square inches.
The radius of a 6 inch round pan is 3 inches. Square this, and get 9 square inches. Multiply this by 3.14 and get 28.3 square inches.
Now, compare the two areas. The 6 inch round is about 1/3 the size of the 10 inch pan, so you could probably make 1/3 of the recipe to fit it. As for the baking times and temperature, I think that you could google up "6 inch cheesecake recipe" and get a good idea from looking at similar recipes how to bake the thing.
I've done the mixing-up-the-egg-in-a-cup thing for years, and generally just eyeball the thing and dump some of it down the sink. Perhaps precision would be better, but I don't measure the eggs.
Generally I increase my 9 inch pan recipes to the 10 inch springform pan, especially during the holidays, by increasing the 9 inch pan recipe by 1/4. Leftover cake is not a problem for us.