I would likely work the cream cheese first, then try adding the liquids from most to least viscous (condensed milk, evaporated milk, eggs, vanilla)
If you try to use a mixer with cream cheese and lots of liguid, it's harder to combine -- you end up getting floating bits of cheese in the liquid (likely what you mentioned as a 'curdled mess'), rather than it fully encorporating. Having the cream cheese at room temp helps, but is still problematic.
It's possible that you might also want to strain the mixture when you're done, as the mixer might leave the eggs' chalaza intact. (if it's too thick to easily strain when done, you can either whip the eggs first and strain them before adding them, or just skip this step)
It depends on how the cheese is made.
Whey will include a lot of the water soluble molecules of the milk. That means a lot of the sugar (and acid if it was fermented) will wash out. Any of the albumin from the milk will also wash out. Almost none of the fat will.
This means that it will be very difficult to calculate exactly how much sugar and protein are left in the cheese but you can get an idea from the type of cheese.
If made with yogurt, for example, the milk should have been heated before making the yogurt to allow the albumin to be bound up and not wash out in the whey. In this case the whey has almost none of the protein.
In many other cheeses, the whey does have a fair bit of protein in it. You can bring the whey to a boil and make ricotta. Weigh the ricotta to determine how much protein is not in your cheese.
As for sugar- If fermented, a lot of sugar is converted to lactic acid. The whey is full of extra sugar and lactic acid but a lot is left in the cheese as well- this is why aged cheeses like cheddar will get more tart over time. The bacteria keep munching on the leftover sugar. I don't know how to estimate the distribution of sugar in these cheeses.
A notable exception is a washed curd cheese like gouda. When these cheeses are made, instead of just draining the whey, the curd is washed in warm water several times. This washes out much of the excess sugar and makes a cheese that doesn't get tart as it ages.
I think the only way to know exact amounts of sugar would involve chemistry skills that I don't have.
Best Answer
Mascarpone is very prone to curdling in mixtures like this if any of the ingredients are different temperatures. Generally, you whip the cheese to soften it and it warms up a bit in the process, then if the milk you added is colder than the cheese, the fats in the cheese will solidify again, causing it to appear curdled.
The best way to avoid this is to make sure everything is room temperature, and to add the liquids very slowly. To fix it once it has already curdled, generally you would gently heat it over a double boiler (or microwave it a few seconds at a time) while whipping it until you've whipped the lumps out. Once it's smooth, let it cool back to room temperature, stirring regularly, and rewhip it once it's cool.