The freezer might work – I haven’t actually tried it, but Gear Aid does recommend storing opened tubes in the freezer.
I usually toss the screw-on brush after the first use, and then I re-use the metal-handled brush even if it’s totally hardened, but in that case I use it more like a spatula. Sometimes I just use my finger to spread the sealant around (while wearing a disposable glove), or the tip of the Seam Grip tube as I’m squeezing it out.
The brush is nice because you can use the bristles to shove sealant down into cracks and crevices, but it’s not absolutely required.
Cheers