Can I charge my battery with a higher wattage charger than what came with device

You can definitely charge it with a higher or lower ampage [but not voltage], so a 5W or 10W (watt) charger will charge a iPad, iPhone or iPod. Note that an iPad will change much slower on a 5W.

This doesn't apply to the iPad Pro (12" model requires a 12W charger)

iPhones and most iPods use a 5W charger. iPad mini and 9.7" iPads use a 10W charger. iPad Pro 12" use a 12W charger.

This is also true for other products by Apple such as the MacBooks. All MacBooks will charge off an 85W charger with the correct MagSafe connector. 13", 15" and 17" MacBooks will charge with a 45, 60, or 85W chargers, however if you use intense processing on a 15" or 17" connected to a 60W or 45W charger the computer will still lose charge but at a much slower rate than if no changer is attached at all. If you don't believe me try exporting a movie from Final Cut or iMovie while on a charger that has a lower Watt rating then the machine would prefer. It will slowly discharge the machine

While all of the above is true, Apple still insists that you should use the correct charger for your device - however it's completely non-essential.

Make sure you have a look at the Apple website as well because they have a recall for the plug part of your charger. They have found that some of them are quite dangerous and can give you a good zap if you aren't careful.

Absolutely. Devices draw only the amount of power that they need, so having extra amps of power available isn't a problem.

The thing you usually have to be careful about is matching voltages

Both the iPad chargers work at 5 volts, so you're good to go.