AWGs produce periodic signals, at least the few I've played with did. The shape of a cycle is arbitrary but the cycle plays repeatedly as the selected frequency. I don't actually know what people typically use them for, other than playing around I never found a need for anything other than the usual sine, square and triangle waves.
Not necessarily. You can run in 'triggered burst' mode where an external pulse causes the waveform to play, either just once or for a programmed number of repeats. Recently, I have been using this to play shaped tone bursts, designed using Matlab to have a specified bandwidth, duration, and autocorrelation function, into an ultrasonic transducer. This is what ARBs really shine at.
The waveforms are only a few hundred points long. I'm using an Agilent 33522B which lets you set the
waveform sample rate to whatever you want (50 Msps in my case) while interpolating up the actual, fixed DAC sample rate of 250 Msps using a digital filter in the internal FPGA.