You're gonna get a lot of different opinions here, and that's ok. There will be some common threads that emerge though probably. Bottom line, go with what seems right and will be comfortable for you. I'll take my shot at them though.
1. Nothing wrong with a 7’6 for the purposes you mention. I’d say that’s about the most commonly used “short” rod for Brookies and smaller streams in terms of length. Some guys so shorter, but the 7’6 will bridge the gap between your 9’0 rod a little better, and will be better on small (but not tiny) streams. On the really tiny Brookie runs, you may at times want a rod that’s a little shorter, but for the most part the 7’6 will still be about ideal and give you a reach advantage over the shorter rods. Truth be told, a lot of times if you can’t make a cast in a tight spot with a 7’6, you probably still can’t make it with a 7’0 or 6’6. FWIW, my small stream/Brookie rod is a 7’0.
2. This is where you will get a lot of different answers probably. Some guys like really light rods (1-2wt), others like a heavier rod (4-5wt). Others split the difference with a 3wt. There’s no right or wrong answer here, they all will do the trick and catch fish. The lighter rods will let the little fish show off a little more and are fun in that respect, but the heavier rods will push line through the brush on a tighter loop. If you plan to fish small streamers too, you may want to look on the heavier end of the spectrum. A lot of times it’s harder to find the 4 and 5 wts in the shorter rod lengths though, where the lighter rods are usually pretty short to start with. FWIW again, my Brookie rod is a 4wt. No matter which you choose, you may want to consider overlining the rod by a line size (I fish 5wt line for instance on my 4wt rod). Regardless of the action of the rod, the heavier line allows you to load the rod with comparatively less line out…this is often useful due to the short casts you make with short rods. There are a lot of people who will disagree on that theory and say you can fish a 4wt rod fine at short distances with 4wt line, and you probably can, but there is an inherent mechanical advantage in overlining a rod for short cast situations, up to maybe 30 ft or so. Above that, overlining becomes a disadvantage, so it should really only be done where most of your casts will be short.
3. Wait, there is no question 3, so I’ll just skip that one.
4. Cast them! Keep your casts short, since that’s mostly what you’ll be doing with them. See how they load in those short range situations, and see what you like. You’ll know when you feel the right one. Cast some different actions…med, med/fast, fast, see what you like.
5. Tough to say, it’s all personal preference. I don’t like the idea of spending a ton of money on a Brookie rod, because it’s gonna be in tight casting situations, and you’re likely to be hiking with it a lot. That said, the following are all about $200 or less and have shorter rod lengths in their line up: Redington (pretty much all of their models), Echo Carbon, LLBean Streamlight Ultra, Cabelas TQR, BPS White River Classic Ultralight, Orvis Clearwater, Cabelas CGR (glass). I’m sure there are a ton more out there. Check the warranty on them, it’s likely you’ll end up using it at some point.
6. Not necessarily, but it’s probably a good idea…I’m assuming you have a fairly decent size reel with 6wt line on it correct? If that’s the case a reel of that size will probably be a little heavy in terms of the balance on a lighter, shorter rod. A smaller reel would probably help. You won’t need anything expensive or with a fancy drag…it’s just gonna hold line 99% of the time and for 99% of the fish you catch you won’t even think about the drag or putting them on the reel. Just something lighter, and comfortable that balances the rod and makes casting easy.