1. Choosing whether or not to show layer controls has nothing to do with the resolution settings. Unless I'm completely misunderstanding your issue...
2. known issues in After Effects CC 2015 (13.5) - fix coming in the next couple of weeks (hopefully)
3. Screenshots please, but it sounds like a bug that should be reported: Feature Request/Bug Report Form
1. Even though I have quarter resolution selected in the Preview Panel, the Space bar invoked preview will render at whatever resolution is selected in the comp window
As you can see above, the comp I stopped (15) is 2:45:00 long and the continuation below (16) starts at 2:10:21 into the same comp but the duration is still 2:45:00 which is impossible.
Definitely file a bug report.
Just out of curiosity, what happens when you render it? Does it end where it's meant to or does it give you extra frames of black?
For #1, show a screenshot of your Preview panel.
For #2 and #3, we have fixes in these areas coming by the end of this month.
When I have my Preview panel set up exactly like yours, I get a quarter-resolution preview when I press the spacebar.