Sounds fun, but if you look at the joules provided by 4 farads, it really isn't that much. I think a single 18650 lithium ion cell has more joules.
Capacitors are getting there. The new ULTRA capacitors with ratings at 100s of Farads catch my eye now and then, but they're physically large, and like you said, need to limit the charge and discharge current, very easy to melt things.
Self-discharge on caps isn't instant, but it's much worse than batteries. Probably days or weeks before most of the magic leaks out.
On board chargers don't tolerate much deviation on input voltage ... 4.7 to 5.1 v is what I'd expect from a 5v nominal lithium-ion charger in a small gizmo. They want at least 4.1v for charging the cell plus a several 100mv for overhead and drop-out voltage.
There's some promising results on the web using 5v buck-boost converters (cheapo from ebay work) with two series 2.7v ultra capacitors (100+ F) The initial 5.4v gets bucked down to 5v, and then boost kicks in all the way down to around 0.8v, so you suck most of the joules out of the caps.
