The iPad uses a 10W charger. The iPhone has a 5W charger. Most older computers only support the original USB spec of 2.5W USB port power support. Your iPad, when on, uses more than 2.5W, so it will not charge unless it's turned off, and when off it will charge at less than 1/4 the speed of the normal iPad charger. That can be up to 16 hours for a fully depleted iPad.

Newer Apple computers, and computers that support USB3.0, support higher USB power standards, but if your computer doesn't then you'll need to invest in a new charger for your iPad.

Note that the iPad will generally charge off other USB chargers from other manufacturers, but only at a slower rate than the original iPad charger, unless the charger you use is specified to charge the iPad. When the original iPad came out 10W USB chargers were not common.

Yes, but only with the iPad in sleep mode.

The iPad requires at least 500mA from a USB port:

Apple peripheral devices may request more than 500 mA (Milliamps) at 5 V (Volts) from a port to function or to allow for faster charging. Such Apple peripheral devices include:

• iPad
• iPhone
• Alluminum Wired Keyboard*

About wired keyboard we get the information:

*When connected to a computer that supports a connection of 1100 mA at 5 V, the first port on the keyboard to have a device or peripheral connected to it that requests standard 500mA power will receive that power. At that point, 100 mA at 5 V is available through the remaining port on this keyboard. This keyboard does not support extra power out of its two ports simultaneously; it requests extra power from the host computer to provide power out of either one of its two ports, then the second keyboard port receives the standard 500mA.

Note: Apple computers and displays that were introduced before 2007 support only 500 mA at 5 V from their ports and do not offer additional power.

[source 1 from support.apple.com]

As the wired keyboards can provide the minimum 500mA required by the iPad (just like older Macs can), it should be possible for it to power the iPad. However, it can not provide an additional current which would allow for faster charging.

But, the new iPad can only charge in sleep mode when connected to a low powered USB hub:

The fastest way to charge your iPad is with the included 10W USB Power Adapter. iPad will also charge, although more slowly, when attached to a computer with a high-power USB port (many recent Mac computers) or with an iPhone Power Adapter. When attached to a computer via a standard USB port (most PCs or older Mac computers) iPad will charge, but only when it's in sleep mode. Make sure your computer is on while charging iPad via USB. If iPad is connected to a computer that’s turned off or is in sleep or standby mode, the iPad battery will continue to drain.

[source 2 from support.apple.com]