Well, either your HDMI port is electrically broken on the board (TV or Laptop, could be one or the other, hard to tell from an email), or it's a software issue.
In linux you can at least have a look at lspci to see if it sees the hdmi port there, that'll tell you if the Linux kernel sees it and whether or not it's a hardware issue. If it sees it, then it's something in the software stack that's not working, probably in Xorg.
In terms of the mini display port adapter, most of those are the cheap passive adapter, which means they can only broadcast a signal out, they can't hear back from whatever device is on the other end. You could try finding an active adapter (they're about 5x as much in terms of cost). The active adapter has a better chance of supporting a full 4k resolution.