I know that using a higher amperage charger will make a device charge faster. But is there a limit to how high the current can be? Could too high a current damage the battery?. Could I use a 24watt (5v/4.1A) USB charger (intended to charge USB-C devices such as my Nexus 5x) to charge an iPhone?. I'd use a USB-A to USB-C cable for connection.
The 6/6S can take advantage of up to 1.6 amps if the USB adapter can supply it. Using a higher capacity adapter will not hurt the phone, but won't help either. An iPad adapter is 2.2 amps, the iPhone "cube" is 1 amp. So an iPad adapter will charge an iPhone 6 or 6S series slightly faster. Not true for the 5 and earlier, which will only use 1 amp regardless of what is available.
It will not "hurt" your iPhone, however with long term use and everyday charging with the larger amperage, you will damage your battery faster and it will start performing worse more quickly than it would normally.
I recommend charging your iPhone with the standard 1 amp charger that is included with your iPhone. However, if you're in a pinch and need some extra battery fast, then using a larger amp charger won't destroy your iPhone.
The iPhone does not need 2.1 amps forced inside it, but it will handle it.
People claim the iPhone is designed to handle more than 1 amp of power, such as the iPad's 2.1 amp brick. Just because there is a cutoff at the amount the iPhone will accept, (not sure what it is just over 2.1 amp I believe) doesn't mean shoving more than 1 amp of power inside it won't damage the internals. The power management chip and battery has to work with more power. (more heat, more wear) Also, using more than 1 amp will increase the temperature of the iPhone overall which is not good for a lot of parts of the phone.(glue, plastic)
Yes, Apple lists that the iPad charger is compatible with your iPhone. It does work, and won't cause noticeable harm for a while. Also, worst case scenario for Apple is you go and drop more money on a battery replacement. More heat and more power is never a good thing, and will cause long term damage and poor battery performance, with the possibility of other things failing faster. I have seen it first hand with my customers before. (I own a small business fixing iPhones) Things such as glue and plastic can loosen up and move. Screens sometimes will get small bubbles under them, screens will sometimes loosen up and require new glue to re secure them into their proper place. The customers that had those issues claimed they regularly charge with an iPad brick.