Where and how should brine be stored?
As cold as possible, in as sanitary place as possible. There's no single right answer. You have to consider the risk of contamination, and how it all fits in with your overall design, workflow, and methods. The biggest threat vector to brine are halophiles (salt loving flora), which include listeria when brine is less than saturated. For extremes of brine management, you can look at a traditional cheese, like parmesan, where brine stays in vats and salt is constantly added, or a modern factory, where brine is microfiltered and UV treated, or filtered and injected with hypochlorite (chlorine).
If in the fridge it seems to take up an awful lot of room.
Then store it somewhere else. Not trying to be flippant, just pointing out there's no single right approach. Weigh the risks and what your inspector will agree too. Obviously, if you never make up new brine, you have to treat it differently (boiling, filtering, storage conditions, etc) than if you make up new batches every few months.
What is the warmest it can be kept?
Are you asking at what temperature will bacteria grow? For biggest pathogen, listeria, 34F stops it. Anything above, listeria will grow unless brine is saturated. But again, this is not to say you need to keep brine at 34F. Weigh the risks and consider your sanitation plan and decide what works for you.
Maybe a better approach is to post your brine management plan and ask for feedback and suggestions for improvement or critique? There's no one-size-fits all. I mean, there is, but it's ridiculously expensive and in almost all cases, unnecessary.