A:AnswerThe streaming stick plugs directly into your tv and requires wireless internet. The stick is powered by a usb cable much like the ones you use on android phones.
The streaming player is a separate unit that plugs into the tv using an hdmi cable. It also has a power supply you must plug in too. The player has an input for a network cable from your internet router and can also use wireless internet too. There is also a USB port to plug in a portable hard drive or USB stick to watch additional content. And finally the streaming player has a remote to play games on the Roku (most of which cost money). The games are definitely not a reason to purchase a Roku. They are basic games and not very good.
As far as streaming content via the Roku channels they are the same.
A:AnswerI'm not sure how you want to use it. I have both my DVR and my Roku connected to my TV, and can use them both just fine. But if you want to somehow connect your Roku to your DVR, I have no idea.
A:AnswerMany universal remotes work with the Roku 3 (though they are IR remotes meaning you have to point them at the Roku 3). It doesn't appear that Roku sells just the remote with voice search yet - you might need to contact Roku to see if they offer a replacement.
