Can I make ice cream with a heavy cream/cream cheese mixture that wouldn't thicken when beaten? Other ideas on how to use it up are welcome!

If you want to make an ice cream, you can try to mix the cream/cheese mixture with sugar, using a 3:1 ratio (cream to sugar). Might have to heat gently to get the sugar to dissolve. Heating would also allow you to infuse any flavors you like, for example vanilla bean, cinnamon, coffee. Chill the mixture. Then churn in an ice cream machine. As long as the cream to sugar ratio is 3:1 (vol/vol), it should produce ice cream with a good texture and flavor. Don't skimp on the sugar, because it affects the texture of the ice cream by controlling the size of the ice crystals. When the ice cream has thickened to a soft-serve texture, you can probably swirl in some jam, syrup or crushed cookies or nuts or candy.

You can also freeze the heavy cream/cream cheese mixture in smaller portions and use it later.

You can also use the mixture to make a pasta sauce with some vegetables, sort of like a primavera or alfredo-like sauce.

I don't see why not....I've never tried it, but it doesn't seem too texturally difficult for the machine. You could add a bit more of a thinner liquid if you're concerned. I vote for brewed coffee, salted caramel or peach nectar :)