Will I be able to browse The Pirate Bay and download torrents?

Yes. Just like you can on your phone or tablet.

Can I use other bluetooth gamepads and remotes or only the stock gamepad/remote work with Nexus Player?

Yes, just like on your phone or tablet.

Edit : kinda don't know actually... But should work since it's the same Android that you can use pretty much any controller with.

Can I transfer content to the Nexus Player via USB? For example, say I have a mp4 on a USB stick, what would be the best/efficient way to be able to get it on the Nexus Player in order to watch it?

Yes, just like your phone or tablet. Although I recommend airdroid. Also recommend MX player, handles pretty much any format although they are hardcore about what devices and os version they support so you might need to sideload the apk.

Edit 2: also recommend ES file Explorer.

I am developing an Android game. Could I technically export an APK, then load it on the Nexus Player and test it on the tv?

Yes, just like your phone or tablet.

What is the micro USB port on the back for?

Just like your phone or tablet, will be for recovery, adb, etc. Mostly will be used for power though, just like the ADT-1.