Is using the two together going to damage my phone in any way?
The short answer is that any USB charger is safe with any USB device; a smaller (iPhone) charger may not charge a bigger (iPad) device. You can also connect your device via a USB extension cable.
The longer version is that Apple's USB chargers (whether sold with the iPhone, iPod Touch, iPad, Apple Watch, or new MacBook) are all essentially the same, the difference is how much electrical current they can supply. The chargers are all regulated (meaning they incorporate circuitry to maintain a constant voltage), which means the connected device will simply draw as much current as it needs and no more.
This means you can easily charge a lesser device with a greater charger; a MacBook (the new USB-C MacBook, not the older MagSafe models!) charger will quite safely charge an iPhone, as will an iPad charger. (You will, of course, need a USB-C to USB adapter, as the charger has a USB-C port and Apple doesn't yet make USB-C lightning cables.)
You can also charge a greater device with a lesser charger, though charging will be slower. An iPad will work with an iPhone charger, but will charge much more slowly, or not at all. An iPhone charger will not damage the new MacBook, but it also probably won't supply enough power for it to run, let alone charge.
You can also connect the device to the charger with a USB extension cable (though not a hub!), but I'd recommend not exceeding 5 m total cable length (from the USB port on the charger, to the port on your device).
Other USB devices can also be powered or charged with an Apple USB charger; I often use my iPhone chargers to charge my 3G WiFi modem, charge Bluetooth headphones, and run a Raspberry Pi.
Not only will it not damage your phone but can potentially charge it faster if you own certain iPhones.
The short answer is: It's debatable. It will not immediately damage your phone, but may decrease long term battery life.
Apple has not officially commented on this, (to my knowledge) however it is commonly accepted that routinely charging a battery with higher amperage than it's designed to take will degrade performance over time. Yet, even that is debated.
How much performance in what time period is not officially known, as consumer smart phones are replaced well before the true limits of the hardware.
Furthermore, anecdotal evidence found online is often along the lines of "I used my iPad charger for 11 months before replacing my iPhone with a newer model" don't really answer the question, although they may for individual purposes.
Here is an article explaining battery basics:
Here is a Reddit discussion with some great information: https://www.reddit.com/r/apple/comments/1uh2zj/is_it_safe_to_use_an_ipad_charger_to_charge/