once you use the cast option, the video will be broadcasted on your tv, while the remote will be available within the slingplayer application > you have just to leave the slingplayer software active on the background while using other applications on your tablet.
As for the firetv I cant tell you if all the remote buttons will be available as I do not have it and most probably i dont have the same set-top box > will be better to start a new thread to find out tte viewing experience from someone with the same set-top as your and firetv stick/box
Here is the solution:
Buy Chromecast and plug it into your TV whenever you will watch. Chromecast supports SlingMedia and it will receive stream from your M1/M2. I have it on two TV's and works every time anywhere.
P.s. And yes, I used to use Slingcatchers myself in past, these are collecting dust now as not compatible with M1/M2 , replaced by Chromecasts. Good luck