As long as you're not overheating now on 2S you can add a couple teeth to the pinion and try that. That will give you better top speed. The trick now will be to slowly start ramping up your pinion size little by little until you find the sweet spot where you are getting the most speed possible...
As I said, fans are mostly there just to cool the motors when there is no airflow going through the body. You can trying the training mode trick and see if that works otherwise just gear it down and/or switch to 2S lipo.
Well if you're running a 3S Lipo you could, it would just be harder to hook up than wiring it to the RX but if you're running 2S lipos you wont have enough power to ran the fan properly, if at all.
Do you have a 5V PC fan? If I remember correctly generally 80mm PC fans run off of 12V so unless you gonna run them straight from the 3S battery then your plan may not work
2S lipos are pretty cheap also. Id just keep the 3S and get a couple of 2S and then you can go back and forth depending on what surface you're running on? Just a thought :)
PC fan in and of itself isnt going to fix the issue. Fans are nice but they put in most of their work when the kit stops moving. Id trying gearing down first, then going to 2S if you can't gear down low enough. Going 2S you may even be able to gear back up a bit afterward if motor is running cool.