It sounds as if you have't fitted the cooler on properly. Or you may dislodged the CPU though that's unlikely.
You may also have knocked the memory or video card out of their slots, maybe only partially, whilst working in there.
Or if your video card has a power connector you may have kocked it off or loosened it.
All of those things could cause the alrm warning.
All you need for the fan controller is a new potientometer (or 'pot' for short). Also known as a variable resistor. You can get them from Maplins and RS and a small one will fit the hole in the Gigabyte front panel. The knob may not fit though so you may have to buy one to suit the new pot, which should be available from where you buy it.
Hold that snapped piece of pot pcb up to the old pot and note where the wires go.
Then cut the wires back and make bare ends of each wire.
Make a note of the value of that pot, I'd guess it will be 5k to 10k, then go and buy your new one. The new one will probably have little metal eyelets rather than a PCB, solder the wires to those the same way round they were on the old, broken, pot.
As for the BIOS put a new motherboard battery in. If you don't have a spare Maplins should sell them, as well as the pot and knob.
If a new battery doesn't sort the problem, the board's probably knackered.