Can I charge my iPad with a non-apple adapter, but with iPad cord?

Yes. As long as it provides the required power to charge. 10w 5v. In other words make sure the adapter is compatible with the iPad.

But you can always buy a new Adapter directly form Apple.

http://store.apple.com/us/product/MD836LL/A/apple-12w-usb-power-adapter

What model iPad? The Ipad Mini uses a 5W charger. The iPad 4 uses a 12W charger. The iPad 1, 2 & 3 use a 10W charger. However, you can use a 12W charger on any iPad model.

 Cheers, Tom 😉