Slow bootup is normal. I rarely shut down my Fire because of this and I don't understand why some people routinely do it unless they just don't know any better (not that there are not valid reasons to do it). A quick press of the power button and it's asleep and wakes up instantly.
Lag and decreased battery time do sound like a badly behaved app. badass battery manager has a per-app battery percentage counter that can be great for highlighting this sort of thing.
As for the screen burn-in (which is not really what it is but we know what you mean -- you can faintly see an afterimage of what was on the previous screen), I've read a few reports of it and the fix is to get a replacement from Amazon. It seems to exist to varying degrees on almost all Fires. I've seen it faintly on mine but not very often and not very noticeably.
I don't play videos on mine but there are lots of reasons a video might abort depending on what kind of video it is and how you are accessing it (streaming from amazon, playing from local storage, etc).