Wow … huh … I don't think I've ever encountered those things before? What version of PS do you have and is it legit? Also what OS do you have and what version of that is it? And for good measure, what tablet are you using?
As far as the brushes, the top settings are your basic ones (mode, opacity, flow, and different pressure settings), if you really want to tinker with a brush, that's when you go to the side menu, it should have a preview of what changing a setting does. You can set a brush to how you want and save all the settings for it as a new brush. (fyi, I'm on CS6 so it may be a bit different for yours depending on the version) Go to the top bar, and in the second drop down menu where it has the brush shape and size click on the little paper icon with the folded corner to create a new brush preset. You can then select this brush anytime you want.
Hand tool glitch: can't say for certain but that might be a hardware software issue? Like sometimes if I'm zoomed in on the canvas and use the brush at a certain size, a mirrored section of the workspace appears all glitchy at the left of my screen and goes away after I zoom out. I looked it up and it's just some compatibility issue with the current Mac OS and CS6. It was worse before I did research and found a plug-in to remedy it. It's fixed for the most part but will still come up on occasion. It's annoying but not "stop the presses I can't work anymore" annoying.
Window mode. Hm, well there's three window mode types and perhaps by trying one or the other it can fix your problem? The first mode is where it's a free floating window and the border takes on whatever border your OS uses. Then there's the tabbed mode, it's attached to the top PS bar and if you have multiple tabs open you can see them all there, kind of like an internet browser. Then there's another mode (you can switch between window modes by hitting "f") where it jumps "into" the window and you can't see the tab anymore. Well, there's also a "display" mode where you go one more step and it just shows your canvas against a black background and all the tools/UI are hidden.
Right hand section: I believe you can move those sections around. Like if they're tabs in the same group you can grab one and move it out of the group and place it above or below.
Disconnect: Do you mean lag? Because it might be another hardware software issue? There's no lag for me when I'm drawing in PS on my Mac, but when I try CSP on my PC I feel it's not as quick and there's just the smallest of lags. I would do some research to try and pinpoint this.