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Yes, you can cut bars down, but there are a few things you need to know. First, if it has a slit in the bars, that slit needs to be recreated after cutting the bars shorter. The slit should be in the same spot and the same width and length. Second, if your bars have a starnut inside the bars, such as for ICS, it will need to be pushed further up into the bars equivalent to the amount you cut off.
The Grit Fluxx comes with a non-integrated headset (it looks like some might even be threaded, which I would argue is not HIC). Integrated headsets should be interchangeable, but I don't know if non-integrated headsets are. They might work, but you should try it out before buying. It's definitely not integrated though.
It might work, but it also might slip from side to side, not staying straight because there is nothing holding it in place other than the one bolt. Some Phoenix brakes had only one bolt, but they had tabs that held them straight within the deck. The best advice is to buy the right brake for your deck.