Sound like you are relatively new to this so here are the basics.
Do you have the correct tools and stands? That will make the difference between the job being a 1 or a 10.
-Impact wrench (electric or pneumatic)
-Chain breaking tool
-Torque wrench
-Stands - I use footpeg stands with my pegs turned upside down so they dont fold or you could try put your rear padock stand through the other side.
Do you have access to the service book? If not here:
Just ordered a couple of the new bikes manuals, will add them as soon as I see them. If any of you have more to add please feel free to share. http://www.teamdouble3racing.com/manuals.php - Site is still partially down and being worked, in the meantime get them here -...
www.mvagusta.net
I use an air impact wrench on the front sprocket nut and it comes off quickly. If you do the same, make sure it is in neutral gear as the shocks from the tool can damage your gears. I wouldn't try the front sprocket nut without something like an impact wrench. Maybe some have come right with a breaker bar, but life is too short for me.
I cant speak for gear ratio's as it is subjective. What do you want the bike to do? Faster top speed with a slower acceleration or faster acceleration with a lower top speed? Your stock ratio is probably fine.
Generally you replace sprockets as a set along with a new chain. I would only change a sprocket if the set were relatively new and for track use. I get the impression yours are worn out, change the entire set with chain.
I recommend this Motion Pro chain tool:
Purchase the Motion Pro PBR Chain Tool 08-0470 from SportbikeTrackGear.com. Get the lowest price, free shipping deal, easy exchanges and no restocking fees - Guaranteed!
www.sportbiketrackgear.com
Sportsbike track gear have a youtube video how to use it.
If you have an older bike (pre 2017) you will probably also need cush drive rubbers as they dont last long. If you have any play on the rear sprocket it normally is because of this.
If you go 520 pitch, you must change both sprockets and chain to 520 pitch. You cant mix them.
I am running a 520 X-ring chain on my F3 and it has been great. That doesn't mean it will be great for you, especially if you do long trips and dont take care of it. Though generally the 520 on a 675/800 is more than up for it. They use them on 1000cc race bikes without issue.
I use: D.I.D 520 ERV3 Chain
Good luck