The candela is based on Watts, but weighted to adjust for human perception. While the "standard human eye" could be considered a physical artifact, we use experimentally determined photopic (in bright light) and scotopic (in low light) response curves, which are supposedly fixed.

It isn't quite as fixed as some measurements since the average human's vision could continue to evolve and change the curve, or because we could find out that we measured it wrong, but it's not dependent on a single physical sample like the kilogram.

What I don't get is why ampere is a fundamental unit while coulomb isn't. I mean, it seems more natural to think of amperes and coulombs per second rather than to define coulombs as the amount of charged carried by a 1 amp current in 1 second.

Techinically, neither the Coulomb nor the Ampere are fundmental. Both are derived from the meter, second, and kilogram.

http://en.wikipedia.org/wiki/File:SI_base_unit.svg

Now, in the abstract, yours is the more natural definition. But experimentally, it turns out to not give the best accuracy.

I believe one starts with (1) the meter (which is based on a second defined using a atomic transition, plus light) and (2) the Newton (which is the force required to accelerate the all-important kilogram at 1 meter per second squared). Then an Ampere is the amount of current flowing through two parallel wires required to produce a force of one Newton per meter between the wires.

http://en.wikipedia.org/wiki/Ampere#Definition

Now, the technology is apparently improving enough that it may be possible to actually count elementary charges like you suggest. See here:

http://en.wikipedia.org/wiki/Ampere#Proposed_future_definiti...

Or not! http://en.wikipedia.org/wiki/File:SI_base_unit.svg