MKV & AVI to MP4 on Windows 7 home premium 64-bit

I know that this will seem strange to post this topic on this site's forum, but I've decided that more people here know what they're talking than anywhere else on the internet. Anyway, I download videos from online, & sometimes those videos are either .mkv videos or .avi videos. I'm looking for a program that will either just convert the container (sometimes called pass-through) or, if necessary, re-encode my .mkv/.avi videos into .mp4 videos with the same resolution. Such a program will need to have the following features.

1) It has to have a version of it that can work on Windows 7 home premium 64-bit.
2) It has to be free (trialware & demoware don't count).
3) It either has to have a GUI, or there has to be a GUI that I can download & use with it.
4) It has to have the option of letting you convert the container or re-encode a .mkv/.avi video to a .mp4 at it's original resolution. In other words, if it's a 1080p .mkv/.avi video, give me the option to convert it to a 1080p .mp4 video. The same goes for any other resolution that the .mkv/.avi video may be in.

Feature #4 is very important to me, because I want the resulting .mp4 video to have the best video quality that it can have, & not being able to transcode or re-encode a video into a .mp4 video with the original resolution doesn't do that.

I play all of my videos on my PC with VLC.

So, what are some good programs with the above features? I've heard of these 3 programs. Are any of them any good for what I'm wanting to do? If a program ends up being good at only converting the container & another is only good at re-encoding, go ahead & recommend them both.

MKVToolNix
XMedia Recode
Handbrake

My reasons for why are these. (I have other reasons too, but not the time to remember them all.)

1) I like my video files to be as small as possible, while also keeping their original resolution. The .mkv format is not the most compressed format out there.
2) I like my video files to be in a format that's widely supported. (I don't just download videos, I upload them too, but the rules here wouldn't like me to go into any more detail on that).

I would think that the first reason would be especially obvious. I understand that videos with X amount of resolution need X amount of file size to store it in, but until there exists a free converter to convert videos to h.265 video format (see the link below), h.264 will be my go-to format for trying to reduce file size while sacrificing as little of the video's resolution as possible. Anyway, thanks for the suggested program.
http://en.wikipedia.org/wiki/High_Efficiency_Video_Coding