Is the alert system still in the main client's code? Will it be removed? It is said that Satoshi removed the alert system from the client ... still, there are at least some leftovers in.

Is the alert system functioning in 0.3.24? Are there plans to remove all traces of it?

See also this answer.
Satoshi didn't remove the alert system; it still exists in the client. However, it cannot trigger RPC safe mode as it could before.

There are no plans to remove all traces of it, as all it does in its current state is display a warning message.