for #1...it's not a temp delay as much as it is a temp "offset". If you've set the temp to 68 degrees with a 1 degree offset (which is configurable). If it hit's 67 the heat will come on, if it hits 69 the cooling will come on. I wouldn't expect it to be frequently switching from one function to another, since you fighting the room temperature which should be consistently above or below the temp you are trying to hold. I would expect it to look more like (daytime) C...C...C...C...C... (then night time) H...H...H...H...H... Especially if you use a themowell so you are measuring the wort temperature which won't fluctuate as much as the ambient temperature.
2.) My 1 stage ranco controller would maintain it's configuration settings even if it were unplugged, so I'd expect the 2 stage controller to act the same way.