I too prefer to do it by ear, whether using guitar or clicking with the mouse. I find (depending on the song) it saves time searching for the MIDI files, and it helps you to develop your own personal listening skills too. Sometimes if you struggle to work something out, it may be a good idea to go to YouTube and search for videos of other people playing the song so you can see where you've gone wrong.
As for the actual remixing of it, I think that (for me) depends on the song. For example, if there's a theme from a game that's already got all the parts added and is a full mix, then I find you have a lot of options, a lot to play around with. Something like that, I find, is easy to turn into a metal rendition (for example), whereas something with a simple melody and not much variation in the original track is more difficult for me to add to and change because I find it trickier to keep with the original style and feel, but for others I'm sure it's more liberating.
Usually, I choose which song to remix based on how it sounds in the first place, how I think I can make it sound, and its complexity. If it's something I want to play on guitar, usually I don't have many problems with anything, but when it comes to orchestrating and MIDI programming (because I'm still pretty new to it), I look for simpler tunes