Would Roku 3 be useable with no internet, to stream from my laptop/Android phone using a Wireless Router?
Would Roku 3 be useable at my cottage to stream/mirror my laptop with Windows 8.1 (and or my Android Phone) to my TV? There is no internet access just a Wireless Router.
Yes??? No? Maybe?? I'm not sure based on your question but I will tell you what they WILL do.
You can stream "moves, music, videos and pictures" from a "USB stick / drive, wired connection to a media server, wireless connection to a media server, the internet".
You *MUST* have internet access to register the product initially. You can't get past the "welcome" screen without the Roku box being activated on the web and tied to your Roku Account. After that, they don't care and you can be "disconnected" from the Internet.
I've never found a way to really "Mirror" the laptop screen on the Roku boxes but you can use Plex to stream media from your laptop. You can use a Roku app for your Android Phone to control the Roku player (it turns your phone into a glorified remote which is actually pretty cool if you have a bunch of them in your house or you need to enter passwords or search for stuff - the phone has a full keyboard) but I've not found a way to display the screen from your android phone on the Roku box. Honestly, that would be horrible. I don't care the resolution of your phone, phone streamed media is poor quality when streamed to a big TV (If you really want to do that, look at Google's Chromecast.)
But, for watching movies that you have stored on your computer, Plex is the way to go. Plex is a free download (open source) for the "Server" portion and the Plex client for Roku (they call them "Channels") is a one time $7.00 fee tied to your Roku Account. (I.E, if you have 10 Roku devices tied to the same account, you only buy the Plex app once). But... downloading "Channels", requires an internet connection so you would want to do that somewhere where you have access.