The amplifier used for headphones is very different than what is used to drive a speaker. It's a totally different component.
It generally isn't, no. Looking at a teardown of the iPhone 5 for example (just the first phone I thought to check) there is a single Cirrus Logic class D amplifier chip that handles both the speaker and headphone outputs. I don't know of and cannot find any phones that have a dedicated amplifier chip for the headphone output although I'm sure one exists somewhere.
Headphone drivers are generally built directly into the audio CODEC because they don't tend to be very large whereas the amplifiers used to drive a speaker tend to be standalone because they drive significantly more power than what you drive headphones with.
In any more modern system, you use a different component or subsystem for each. iPhone 5 is a pretty old example and speaker technology has come a LONG way in phones. These days, you're driving watts of power into your phone speakers, any headphones driven with that much power wouldn't last very long.
Even very high end headphone amplifiers (external, chunky ones) tend to be rated for around 1W max. Typical is much lower, in the 100mW or less range. Either way, you aren't listening at that high of power unless you want hearing damage.
It generally isn't, no. Looking at a teardown of the iPhone 5 for example (just the first phone I thought to check) there is a single Cirrus Logic class D amplifier chip that handles both the speaker and headphone outputs. I don't know of and cannot find any phones that have a dedicated amplifier chip for the headphone output although I'm sure one exists somewhere.