The RF that different products use generally isnt compatable, unless you are talking bluetooth, so even if you get RF tivo remotes, they wouldnt be able to talk with your TV's via RF.
That said, if you do it right, the remote should both turn off and on your tv. There are threads on here about your exact problem (turns off tv, but not on... or vice-versa)
-You may not be actually using the "perfect" IR remote code for your particular hardware.
-the Tivo remote can actually send out more than one IR power signal when you hit the power button so sometimes you have to program two codes in there to get it to work with hardware that uses two different codes for on and off.
-There are "learning" tivo remotes that can learn remote codes, such as the Slide Pro-the "current" one that they sell.... it has the slightly revised button layout that the Roamio remote uses and a snazzy keyboard inside. Still available out there in the wild are "glo" remotes which are backlit (and have the original button layout/size that match the Mini's remote)
If you get a Slide Pro and get the one that comes with the RF receiver(its the same price), you can use the RF receiver with basic RF remote that comes with the Roamio, with a mini. It doesn't have the IR learning capability, but you dont need line of sight (e.g. under blankets when its cold or you can hide the mini away, etc).
Streaming to the IOS app is still a "young" feature and not matured. We are hoping that one of the "neato" updates that are promised to be coming in the summer update is both quality improvements as well as long-promised Android support.
If you wanted to troubleshoot the Moca situation, you might want to put that in its own thread.... that is a discussion onto itself