Do you know if I can burn movies I downloaded using my external blu ray/ dvd writer
That depends on the file format that the downloaded movies are in. If you refer to normal purchased movies, then, no. You would only be able to burn them on a disk as data. They would not be playable in a DVD player.
Mac does not have built-in support for Blueray, only DVDs.
A more general, question... Why do you need to burn them onto a DVD disk? If you have the files on your iMac and want to watch them on your TV, there are ways to do that. Via Apple TV is one, connecting via HDMI cable or Wi-Fi are others. To watch your videos on your iPhone, use iTunes to copy them to the device. If it's home videos that you made, keeping them in iCloud Photo Library would be an easy way to have access to them, without permanently taking up storage space on the phone.
http://www.apple.com/tv/