If you want to condition in the bottle, you only need enough room in your fridge for the beer you want to drink.
Natural carbonation/bottle conditioning happens at room temperature (around 70F). You add a sugar solution to the beer at bottling time, cap the bottles, and leave them at room temp for 3 weeks or so. The yeast nom noms on the sugar, and creates CO2 to carb your beer.
But as you note, CO2 more readily dissolves into cold liquids. So much of that CO2 is in the headspace, now under pressure. This is why you chill the beer overnight before tasting: it allows the CO2 in the headspace to dissolve back into the beer. If you just throw it into the fridge for an hour, the beer will be cold, but the CO2 won't have had time to dissolve into solution. That's when you get a ton of head and flat beer underneath.
As for flat force carbonated beer, the mechanics are the same. When I force carb, I chill the beer to 38F, then apply 12PSI for 1-2 weeks. According to a force carbonation chart like this one, that gets me to around 2.6 volumes of CO2.
To get that same 2.6 volumes of CO2 at room temp, you're off the charts for PSI. At 65F, you'd need to keep 30PSI on the beer for the same amount of time. So for most folks, it really isn't feasible to force carb at room temp. Or really, it's just a lot more convenient to force carb in the fridge, since you can use a lower CO2 pressure, which your regulator is probably already set to.