Yes
improving

Yes. Rating fluctuations are common and especially when starting out, it sometimes takes a while to stabilize rating to your actual ability, so rating may even fluctuate by a few hundred points. Rating is always going up and down, but it tends to change slightly less dramatically once you've played for a while and your rating stabilizes to your current chess ability.
keep going from 800 down to 600 and sometimes even to 500 is this normal