April: Aida has already answered, but I'll answer too: yes and yes. And I even go for a run three mornings a week, then come home and shower and shave my pits (with soap) and it seems to stay on fine -- and is totally invisible. It will have worn off by the second day, though.
They have a morning roll-on deodorant that you can use for extra protection if you want. I use that if I know it's going to be a hot day, or if I'm going to be doing anything strenuous. You still use the nighttime roll-on, but you can put a bit of the morning roll-on after your shower (and shaving) and it won't irritate.
Word of caution: DO NOT apply it after shaving. I did that once and it really stings! But luckily I didn't develop any redness or anything.
Aida: thanks again for the tip! You're awesome.