I am a female with HSV 1 genital since 6 years (was swabbed and confirmed) and I have never had a cold sore. I have been kissing a new male partner and he is very worried about transmission. Can he get my HSV 1 through kissing me on my mouth or drinking from my bottle? 

If your infection is only genital, there is no risk of kissing or drinking from the same bottle.  None---

Thank you! As a follow up question, if my partner has had cold sores in the past, would that mean he is not at risk from getting genital HSV 1 from me?

That's correct.  The risk of transmitting HSV 1 to a new location on your body is extremely low.  I never say never, but it were me, I wouldn't give it another thought---

Finally, how great is the risk of my HSV 1 being transmitted to my male partner’s genitals if we were to regularly have protected sex? (Both scenarios please… #1 he has HSV 1 already on his mouth… #2 he has never had a cold sore, so no HSV 1 potentially in his body). Thx!!

If he has HSV 1 orally, the is very close to zero risk that he would now acquire this genitally from you.  If he doesn't already have HSV 1 in either location, there is a very small risk that he could acquire HSV 1 from you through intercourse.  It has been documented in the literature, but I've never seen it.  It is far more likely to occur when the person who has genital HSV 1 has just recently acquired it vs a few years down the road.---