From your first problem... I'm not sure the key 'can' be removed and leave the ignition switch on. But maybe I'm wrong.
So, I guess the problem could be in the ignition switch, but that's a stretch.
With the R3, the normal answer is a bad or undercharged battery OR a bad connection at the starter relay.
You have lights, so the main 30A fuse is good.
There is occasionally a poor connection at the starter relay where the main 30A fuse and its holder connect. Signs of darkened metal or melted insulation at the connections is a pretty good indication of a loose and overheated problem that needs to be addressed. (It's easier to remove both front and seats to get into here and work.)
Kill switch is in 'Run' position? Bike is in Neutral?
Do you hear the fuel pump priming in the fuel tank when the ignition switch is turned on?
If all above is good... pull in and hold the clutch lever while trying to restart the bike.
If the engine turns over and starts properly... the kick-stand safety switch may be the problem.
Pulling in the clutch by-passes it.
If all the above still looks good... lights are still on, gauges still light up, but the engine still won't turn over... Fully recharge the new battery and try again.

 

Hello all,

Typical first post, my bike won't start. Here is a recount of events:
I stopped to get gas, when I took the key out and filled up, I realized the bike's ignition was still on. After turning it off and waiting about a minute, it did not start. Battery was giving power, lights were on, but when attempting the starter, I would hear a click near the 'starter relay area' and the lights would flicker and turn off. I bump started it and rode it home roughly 40 minutes afterwards.

I turned it off and let it cool down and afterwards, the battery was dead. I replaced the battery and there is now power and lights but the bike still won't start. I checked the battery, terminals, connections, and all fuses. I am better with cars than bikes so I am not entirely sure where to go from here. My starter relay seems to be in good condition but I am not sure. (Fuse is good and no signs of overheating or external damage). If there are any tips on what to check next, I'd greatly appreciate! Thanks!

I highly suggest searching the forum, as this has been addressed extensively.

A new battery does not mean it's fully charged. Often you need to charge the battery. There are many things to do before you ever get to replacing a battery as well, and first and foremost before ever starting to diagnose electrical issues you should have and know how to use a multi-meter and have a service manual on hand.

@airhead83 has already addressed great things to check. However, without a multi-meter, what are you doing replacing a battery? Do you have a battery charger? Get a multimeter, check battery voltage properly. It should read 12.8v minimum with the power off, key out. 13.1v minimum when running at 5k rpm.

 

I don't think you can take the key out if it's on, so your switch is broken. But doubt that's the issue. Sounds like battery or starter relay to me.