As fast as you like.  Just like a petrol powered car/bike, you can modify and tune your EV to reach optimum levels of performance. While most EV’s are commuter cars, there are a few production cars that offer great performance. 
The most talked about would be the Tesla Roadster; an all-electric sports car with a 0-60 mph time of 3.6 seconds and 245 miles of driving range between charges. 

On average, an EV with the older lead acid batteries or valve-regulated lead–acid battery packs or the new Lithium Ion packs will charge fully overnight.  The Mahindra e2o EV hatchback charges in 5 hours.  The Hero Electric E Sprint electric scooter charges in 8 hours.
With Lead Acid batteries, 70% of the charging is done in the first 40% of charging time. The rest is the “topping up” stage of the charging process. Technology is changing constantly however, and with new battery technology comes faster charging times.  
A possible solution – the ability to completely charge an electric car in a matter of 20 minutes. This is made possible thanks to the utilization of a single standardized connector which will accept DC fast charging.  
Speak with your EV manufacturer if there is a provision for DC fast charging in your EV.

Not for many years. Mahindra offer warranties covering 5 years of driving on the lithium-ion batteries in the e2o.
Reva users have used batteries for more than 3 years.
One simple technique that could be used to pay for the battery is to start an Recurring Deposit of Rs2000 per month for 3 years or more.
Liquidate the RD after 3-4 years whenever the need arises to change the battery. Thus bank interest accumulation can partly help you fund your shiny new battery pack! 🙂

For all the energy required to propel a vehicle, not all of it makes it to the wheels. Some of it is lost to friction and heat. Vehicle inefficiency can be classified into two categories of losses: road-load and energy conversion.
In an electric vehicle, chemical energy is stored in a battery. Lithium-ion batteries are used in EV's because of high energy density. Converting the chemical energy to free electrons (electrical energy) can be greater than 80% efficient – some energy is lost to heat in cells and other battery pack components such as current conductors and fuses. Overall, drive efficiency of an EV more than 80% - almost three times more efficient than an internal combustion powered vehicle.
When you compare this to Internal Combustion Engine vehicles, chemical energy is stored as gasoline in a conventional vehicle. Combustion is used to convert the chemical energy into thermal energy. Pistons convert the thermal energy to the mechanical work that turns the wheels. The conversion process is, at best, 35% efficient. The majority of the energy stored in the gasoline is lost as heat.

Exactly how much will vary depending on the vehicle and electricity rates. On average, it will be less than Rs 40 for an all-electric car or a scooter.  Your overall energy bill will be lowered by driving with electricity.
For example, the Mahindra Verito gives a mileage of 15kmpl.  At Rs 75 per liter of Petrol, the car would need 4 liters to cover 60 kms for a total cost of Rs 300 for the trip.
The Mahindra e2o would need 10 units of power for a full charge of 100 km. At Rs 4 a unit that would be Rs 40.
So there you go, a person driving an EV spends less that Rs 30 to cover 60kms while a gas powered car would really drill a hole in the wallet. On the other hand, if you install Solar Panels, you can run your car on the power of the sun for free.

Lots of reasons actually -  
The manufacturing scale high-output, high-capacity cells is relatively small. There aren’t the same demands for lithium technology from shops, industries and the auto industry, as opposed to lead acid batteries. This means higher costs.
Then the fact that you have to buy an often-expensive Battery Management System to look after your lithium batteries puts the overall purchase price even higher.
While a shortage of available lithium plays a part, it’s not a big part. The amount of lithium inside each lithium battery is surprisingly small. Couple that with lithium being in great abundance – despite what you may have heard – the only thing hampering it’s supply is the limitations of the existing lithium mining industry.
The next main reason why the batteries are so expensive is because they are very complicated. Each large lithium cell takes a considerable effort to produce. Both chemistries and manufacturing conditions must be absolutely precise. Special manufacturing facilities are required, which come at a price.
So in order for the costs to come down, people have to buy more and create more demand, which will create more manufacturing, lowering prices. The cost of the batteries are dropping quiet rapidly.  They have dropped in half over the last 4 years and experts say that the costs of the batteries will drop another half over the next 4 years. So when the cost of the electric vehicle batteries have dropped enough , the incremental cost of an electric vehicle will not be that much higher than an oil based vehicle.

Most people recharge overnight in their own garage, or in an apartment basement, or driveway, but there are public chargers for electric cars as well in parking garages and shopping centers.  Contact your EV manufacturer on public chargers.

Putting solar photovoltaics directly on EVs is nice but not adequate. Most solar panels would add too much weight to an EV to be worthwhile. Some newer, lighter, flexible PV technology could generate power for interior climate control or minor tasks, but not enough to power a car a significant distance. We need to have that plug.  Windmills on EVs don't make sense. The drag they create reduces efficiency, necessitating more energy to run the car.

Hydrogen vehicles are the "car of the future," and probably always will be. Hydrogen fuel cell vehicles (FCVs) are EVs, but instead of getting their electricity from batteries charged from the grid, they get their power from fuel cells using hydrogen as the energy carrier. The tricky question is, where does the hydrogen come from?
The reliable and steady source of hydrogen will have to be a hydrogen pipe line coming from a hydrogen purifier company. This is impossible for now because we do not have hydrogen pumps at local petrol pumps or hydrogen pipe lines coming into our houses. It will also take billions of rupees of investment and a long time for all local petrol stations to have a hydrogen pump.
Who will pay the billions required for this new infrastructure? 
With plug-in EVs, the infrastructure is already in place - the electric grid.