How to improve one's bike handling skills? Are there any time tested ways to improve one's bike handling skills? 

Outside of regular riding (road) are there certain exercises one can do to increase body awareness on the bike, improve coordination, and increase balance?

Are there any exercises one can do off of a bike to increase handling skills and balance while on the bike?
In rapidly descending order of importance ...
(see Why I chose these priorities below)


Practice track stands. See How to do a track stand?
This teaches balance and slow bike handling skills. Also practice riding in confined spaces (the last time I fell off was trying to do a U turn on a path that was 4 ft (1.2m) wide :-)
Learn to jump both wheels off the ground at the same time. This is usually called a Bunny Hop. Apply to jumping objects.
You probably already do simple bunny hops to get up kerbs. See our bunny hop database https://bicycles.stackexchange.com/search?q=bunny+hop
Learn to pick objects up from the ground with your hands (on both sides).
Have you or a friend ever dropped something on the road and had to circle back, stop, and pick it up? With this skill you learn to control the bike while hanging off the side, with your knuckles dragging ...
Start by placing a drink bottle on some level ground, and ride slowly past it and pick it up with one hand. When you can do it with one hand, use the other. Then go for smaller objects. Last week I picked a pump a guy ahead of me in the bunch had dropped. Be very careful with long objects; if they get in your spokes you'll never even know what happened :-)
Learn to mono. Ok, so I still haven't grown up.
Learn stoppies (mono on front wheel). Just how much can your front brakes cope with?


The first two have obvious utility, the third is useful (but 99.9% of cyclists live without it) and the last two are more for fun. But they will all teach you balance and control.

But probably number zero should be learn to ride with no hands. Of course, this is illegal to do on the road, and of course every kid does it. Once you can do it well, you can put on and take off jerseys, rain tops, etc. Being winter here now, this one gets used every week.

After a bit more thought, I would add a couple of easy skills that teach a lot


Learn to initiate a turn using only your body weight. Most people turn by countersteering a tiny amount to initiate the turn. This skill teaches you how the geometry of your bike works, and links in with balance skills.
Practice riding along the painted road lines and see how far you do it. The further you can go the more balanced and fine tuned you are.


And seeing the answer by dlu reminded me to add 


On fast sections, where the surface is unknown (or known to be bumpy), get your bottom off the seat so that the bike can pivot under you. 


Why I chose these priorities

Items 1 to 5 are ordered as a combination of potential frequency of use and difficulty / danger involved in learning. Certainly @mattnz is correct that stoppies are valuable in emergencies (being able to do one saved me from a HUGE T-bone once).