How much brine do I need to freeze icecream?
Brine

Some assorted thoughts, before I try to guess an actual quantity:

Dependence on overrun shouldn't really be significant, as long as you're mixing well. The air doesn't add significant heat capacity, but it will make heat transfer less effective by providing some insulation, so you do need to mix well.

I can't imagine dependence on type of ice cream being that huge - whether you're using cream or egg yolks, you've still got plenty of water. I found a table for specific heats; it looks like eggs have a slightly lower heat capacity above freezing, and slightly higher after. I'm not sure about just the yolks, or about the latent heats of either. In any case, the properties of the ice cream custard will be somewhere between water and those. (Wildly generalizing, ice cream might be 1/5 sugar and 1/6 fat by weight.)

The more salt you have in your brine, the colder you can get it before it freezes. You won't be able to get it as cold if you don't have as much salt. I think you were slightly misinformed in chat: 0F (-17.7C) was defined (for whatever reason) using ammonium chloride. Saltwater brine (sodium chloride) can reach -21.1C (-6.0F) with 23.3% salt by weight (source: wikipedia). I believe the freezing point is roughly linear as a function of salt concentration; in any case, it'll be the most effective with the highest concentration.

You missed one factor that may well be bigger than anything you listed: heat coming in from the environment. This will depend on how well insulated your outside vessel is, the ambient temperature, and how much surface of the brine is exposed. (Also on how much you're gripping it with your hands, I suppose.)

As for quantities, I could try to ballpark something using specific heat/latent heat, but I think a far better method is to just look at the recipes for "ice cream in a bag" - the general idea of those is to put the ice cream mixture in a smaller plastic bag bag, inside a gallon bag full of icy brine, and shake. There are a lot of recipes using a sandwich or quart bag with 1/2 to 1 cup volume (probably ~120-240g) ice cream mixture, inside a gallon bag with at least 2-3 cups of crushed ice plus plenty of salt (maybe 300-600g? hard to say). I also found this one with the brine just in a bowl, using a pint (probably ~475g) of ice cream mixture and 3-4 pounds of ice. The latter is a little easier to estimate based on; it suggests for 100g of ice cream mixture you might need 300-350g of brine. (The other recipes are consistent with this.)

Mixing

This is really a separate question from all the brine stuff. If you want to get good ice cream, you absolutely must mix (churn). This both helps minimize crystal size and incorporate air. (Unlike whipping a liquid, mixing during freezing doesn't have to be as vigorous to get the air in.) It's possible that if you freeze it fast enough, you could make up for the air part with your pre-whipping, but I suspect you'll lose a bit of that volume during freezing. So at the very least, stirring will break up crystals, and it may well give you airier, fluffier ice cream too. (I'd be careful with that New York Times recipe I linked to; it's going to be prone to giving you unwanted crystallization if you don't take care with the "kneading", and may not leave a lot of overrun.)

General advice

I know you may have reasons not to do this, but in general, if you're willing to go to this much effort, you might as well buy an ice cream machine if you can. The part that you freeze will probably take up less freezer space than the ice/brine you'd otherwise use, and it'll also do the churning for you.