Baking – Do extracts add sweetness or just the flavor

To increase flavor extraction (and this applies to ALL flavors, not just ginger):

• Simmer for a prolonged period (more time to extract flavor).
• Puree or finely mince the ginger, then strain it out with a fine chinois or cheesecloth. Smaller pieces allow water to more easily penetrate, and allows flavor compounds to be extracted more quickly
• Stir frequently; this greatly reduces the time you need to cook the ginger, by ensuring flavor compounds reach an equilibrium throughout the mixture, rather than being concentrated at the surface of the ginger pieces.
• Cook the same ginger multiple times with fresh water each time. This will extract more flavor than doing one batch cooked 3x as long, and also more than cooking the same ginger with 3x as much water. Each batch will be less flavorful, of course, but after 3-5 batches, you should have most of the flavor out.
• Reduce the ginger water after extraction to concentrate the flavor. This will be especially helpful when combined with multiple batches.
• Add alcohol to the water, as many flavor compounds are oils, and thus more soluble in alcohol than water. This is excellent to combine with reduction, as the alcohol is easily evaporated.

All of these approaches are based both practice and the chemistry behind extractions, which is what you are doing here.

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)