A few thoughts whilst reading through the thread...
At least in the radio-control world, where my experience lies, large electric setups running at up to 5 horsepower or so are now quite common. A typical setup for an electric helicopter may have a 500g motor putting out about 4 kW. Over the past year or two they suddenly came of age. They're clean, reliable, quiet... last time I showed up at a flying field there were no IC engines in sight. The only maintainence they need is to lubricate the bearings - though motors to drive large propellers may be lighter and more efficient if they also incorporate a gearbox, which would obviously increase complexity. The main failure modes are for the bearings to give up, to shed rare-earth magnets which are epoxied into place, or if they're overheated then the magnets weaken.
Brushless motors from reputable brands are very reliable - unfortunately the same can't be said for the speed controllers (ESCs) that drive them. If they go wrong, they have a nasty habit of starting battery fires and burning out expensive motors. They often cost more than the motors they drive, and judging by the number of product recalls, designing a reliable one can't be nearly as easy as it looks. The basic circuit involves bridges - i.e. two switches in series which, if both turned on at the same time, will short circuit the battery.
Cooling is a perfectly tractable problem. One of the great advantages of an 85% efficient motor is that it generates much less waste heat than an ic engine of equivalent power output - you don't need to remove 100kW of heat from a 100kW motor as someone mentioned previously. Most model aircraft motors incorporate centrifugal fans, though I'd wonder whether these would still be effective on a propeller driven aircraft operating at a much lower RPM.
Rare-earth scarcity is going to be an issue in the short term. It's already pushing the prices of brushless motors up quite considerably. Prices for small motors have just about doubled over the past 5 years. However, new mines will come on line in the next few years, and there are research projects to develop magnets based on the commoner rare earths. My guess is that this will be a much bigger issue for cars than aircraft where cost will be less of an issue, and weight more important. At present, a 5KW motor costs about £150. Lithium scarcity (for the batteries) is another potential problem.
I don't really know enough to comment on whether electric aircraft will cause problems for electricity grids. I suspect not - for pleasure flying, overnight charging will probably be practical. Airfields could have on-site generators. Batteries in aircraft that had been charged overnight but were not currently in operation could be used to charge other aircraft by day. Microlights that only get used once a week for an hour of pleasure flying could possibly get by with a big solar panel.
Electric motors are different from IC engines. They're physically smaller for the same power output. After a certain size, big motors aren't much more efficient than small motors, and two smallish motors aren't much more expensive than a single bigger motor because a lot of the cost is in the materials - they're mechanically much simpler than IC engines of any variety.
Batteries are also different from fuel. They don't get lighter as you run them down. On the upside, this means that their center of gravity doesn't change and their placement in a new-design aircraft can be more flexible. We've talked a lot about energy density, but something that has only been touched on is that power density for an electric aircraft is potentially much greater than for an IC one - even a jet. My model helicopter's batteries can supply 60C at 36v at a weight of 1kg and a total capacity of 4 amp hours. In other words, about 10 horsepower per kilogram... for a whole minute. A 10 horsepower motor will weigh another kilogram... so all in all we get about 5 horsepower/kilo. 200 kg of batteries and motor... 1000 horses, instantly available at practically any altitude. This might not be sensible, but in between this hypothetical aircraft and the electric motor gliders currently being built, there may be some rather interesting compromises.
So, whilst in many cases it might be straightforward to bolt them on to existing airframes, aircraft designed for electric power from the outset will probably look quite different from existing models. You could build a twin or triple with very minimalist engine pods that cost very little more than a single. You could perhaps build a 500kg ducted fan aircraft that would outfly most military aircraft, for a while minute. An electric VTOL jump-jet. A simpler gyrocopter with one motor to do the pre-spin, and a separate motor for the propeller. With regenerative charging of the batteries, you could slope soar for an hour or two and generate enough power to do a long cross country flight.
Batteries, unfortunately, are the key. Lithium polymer batteries only last a few hundred charges, even when you take good care of them. A123 lithium-Fe batteries are much better and can last thousands of charges, but weigh about 20% more per unit of usable energy (if you discharge LiPo past 85% of rated capacity their lifetimes are shortened considerably - LiFe are much more forgiving). Lithium polymer also burst into a ball of flames if you puncture them, but you can drive a nail through an A123 battery and it just gets mildly unhappy. The last I heard, Lithium air batteries had the capacity, but not the discharge rates required for sustained flight, but I haven't looked into it for a while. Anyway, if we ever do get a battery with the capacity of Li-air and the robustness and safety of A123, then things could get quite exciting, though I agree it's likely to take a little while for the changes to percolate through.