Hi and welcome to the forum.
Many of your suggestions sound familiar as they have been requested before, though not exactly in the form you suggest. You will probabably find helpful workarounds for many of these problems in earlier forum posts.
The ability to re-order children by dragging the children towards the bottom of the screen and letting go move that child to the beginning of the (di)splay.
A mouse-driven way to custom order Thoughts like this should appeal to the developers.
Can I suggest a spider web view or splash view
Expanded View will allow you to position Thoughts like this. In fact, if you have a dozen or more child Thoughts, Expanded View will space them out radially at a distance from each other, like a spider's web.
Having a 'master brain' that can be toggled on/off to get at all other "thebrains" files
Linking two brains together is one of the most popular suggestions. Look in UserVoice.
A general simple notes drop-down is needed
Notes and other tools can be auto-hidden and reappear in the Pro Edition.
And to then have the ability to jump through the children using the tab.
I suggest trying Outline View for this and using the cursor keys. However, I agree that Normal View does not make it easy to progress through all the children by pressing only one key. Another solution is to display only the children in Reports by tagging them, and stepping through them that way. I don't know if it's my screen or not, but the font you've chosen seems awfully large. Perhaps you could go back and reduce its size? Keep posting.
PB 18.104.22.168 on Windows XP, J-1.6.0_17