These Yamaha's are bulletproof. There was a 2009 FZ6R on the local craigslist with 230,000 miles on it - it looked a bit ragged, but supposedly still ran fine.
I know of a 2002 R1 with 89,000 miles on it and it still runs like a champ.
I think you're looking for Part #3 on this Parts Diagram: https://www.bikebandit.com/oem-parts/2013-yamaha-fz6r-fz6rdl-side-cover/o/m155037sch779289
Bike Bandit is always my go-to source for OEM parts - great prices and they are pretty accurate on delivery times.
Before you spring for a new battery have the old one checked out, it may still be good. Definitely get yourself a smart charger, like a Battery Tender. A properly maintained battery can last a long time, the last one in my R6 lasted 9 years.
Not knowing how much of a charge the original owner...
These engines are pretty much bulletproof. The valve inspection spec is every 26,600 miles and it really doesn't need done before that. 10k miles is nothing, don't waste your time doing a valve inspection now. Just go ride.
I run Pazzo Racing adjustable levers on my R6 - a standard length on the clutch side and a shorty for the front brake. That combination works great for me and I plan on installing a similar setup on my FZ6R.
OP, instead of worrying about how low you can get in a corner, you should concentrate on getting through the corner smooth and efficient. As you gain experience your lean angles will increase. That being said, please save knee dragging for track days.
Regarding tires, the stock tires are more...