Is it possible to add the Bluetooth profile of A2DP to a phone that does not have it?
Thank you for any clarification you are able to provide.
What are Bluetooth profiles considered: software? hardware? firmware?
A little of each. All of those levels are involved. A profile is really just a way of describing set of functionality; each profile requires some support at all levels of the Bluetooth stack to work, right down to the hardware.
...Is it possible to add the Bluetooth profile of A2DP to a phone that does not have it?
Generally not, although there are rare exceptions.
The hardware (and usually firmware) need to already support the missing profile. It needs to be a situation like the software developers left it out due to time constraints, or disabled it because they weren't able to test and tw...