You should stick to models unofficially supported by Sony (list here).
Anything else will surely fail because of the way the commands are hardcoded/remapped/reserved in the firmware to fit the Sony TV remote.
Sure, there are plenty of apps to remap them, but they will not work due to the lack of root privileges.
If you have a PC with a recent Nvidia GPU you can always combine Moonlight with VirtualHere USB so as to bring the PC screen via LAN to the TV and use an unlimited number of controllers.
As for the "Can I connect 2 gamepads and a keyborad-mouse all together at the same time via bluetooth?" part. Provided the controllers are part of the above list (recognized by the TV according to their HIDs) I am pretty sure you can use as many as you say. Also, so as to preserve the USB ports, remember you can always attach an USB hub.
Personally I have a keyboard and a gamepad (wireless) connect to one USB hub, which also offers the advantage of a better placement/improving reception (for instance, I found that a placing the USB receptor in a port where it is was obstructued by another USB device - say a storage pen - prevented reception).
