The AI is itself a program. If the AI can write programs, it can write a better version of itself. A few iterations of this, and it will have far surpassed human intelligence.
So it would be more accurate to say that writing programs is the last job that will be automated.
Not necessarily (or even likely). There's a huge difference between being able to write "programs" (for example the ones described here are ~5 lines) and being able to write a large and very complex piece of software, which such an AI would undoubtedly be.
There's also the question of what constitutes "better." Improvements in e.g. speed or memory efficiency (hard enough as it is, but conceivable for an AI) are a quite different thing than improvements in capabilities or understanding.
That's true. real-life programs that programmers get paid to write are typically more complex than 5 lines. Any AI good enough to replace real programmers is going to be very close to good enough to rewrite itself.
I would think it would be much more likely follow a normal evolutionary model - improve itself to a local maxima, then just stall while the random mutations fail to make the huge leap required to hit the next maxima.
We can't even properly model the human mind in computers yet - not enough processing power - so it follows that an AI which is "better" than humans would require even more processing power than that required to ape humans.