Glad you had a nice ride on the way home from picking up your new to you bike.
The low fuel light typically comes on about where yours does. 23 liter tank, still usually 6 liters of useable fuel in the tank when reserve comes on. Gives you plenty of warning to get fuel.
You don't say how many kms your new to you bike has. Gen I Super Tens, (10-13 model years), suffer from vibrations under load in the 3-4k rpm range as they get higher miles on them. Some simply don't notice it. Others start to notice it somewhere between 50k and 80k kms. The solution is to change to the Gen II clutch hub, (Primary Driven Gear in Yamaha speak). It uses polymer bumpers instead of springs and eliminates the vibration under load from the worn clutch Primary Driven Gear. This isn't a super difficult job, but requires a clutch tool and impact wrench along with a torque wrench to re-tighten the nut. The clutch cover gasket often can be re-used, but may tear. The nut can be re-staked, but is supposed to be replaced with a new one each time it is removed. The following parts would be ordered to do the job:
Primary Driven Gear Comp - 2BS-16150-00-00
Nut - 90179-20007-00 (can sometimes be re-used)
Ring - 2H7-16385-00-00 (can sometimes be re-used if care is taken during removal)
Gasket, Crankcase Cover 2 - 23P-15461-00-00 (can be re-used if cover comes off w/o damaging gasket, which is often the case)
These parts cost approximately $350 USD plus shipping. The main cost is the Primary Driven Gear at approx. $300 USD.
Nothing bad will happen if you don't change the clutch hub to the newer design. It's very rare for them to fail, it just has the vibration. I did change this on my '12 at around 133k kms and felt it was a big improvement.
On the bar wag on washboard gravel road. This could be something as simple as the steering head bearings needing to be re-torqued. It's common for them to come loose over time and wear. To do this right it requires an inexpensive tool and a torque wrench. There is a process for doing this that requires torquing the lower spline nut to 52 Nm, then loosening it slightly, then re-torquing to 18 Nm. The top hex nut under the rubber plug on top of the upper triple tree has a torque spec of 130 Nm and is commonly found loose on new bikes after some break in miles. If it hasn't been addressed, this could also be the case on your new to you bike.
If you do this or have it done, it's a good time to clean and grease the steering head bearings too. A little more work, but worth while. Yamaha tends to use minimal grease throughout the bike. I believe there are some write ups on the forum on doing this service.
Hope that helps.