I know that the snow can be boiled by setting up a campfire outdoors, but rather than spending additional resources and to generate more heat I may want to use the fireplace at my resting place. Does the game allow collecting and carrying snow with you?
Snow isn't carried as part of your inventory, it's simply abstracted as the ability to make water at literally any fire.
The closest thing to this - and it's of limited usefulness - is, basically, parboiling water: You can melt it to non-potable water, which can then be boiled to be safe to drink later on; you can also then purify it with purification tablets. The only real use for this is if you've already got a fire going, but feel the need to relocate and you're allergic to wasting firewood (like I am). You can use up the committed fire energy making non-potable water, and then finish the job elsewhere.
Water Purification Tablets let you double your water production per unit of time and/or firewood in this way since melting the snow takes the same amount of time as boiling the non-potable into potable.