There are no documented cases that I am aware of of people transmitting HSV 1 (or HSV 2) through sharing utensils or cup. I would say that chance does increase if someone is showing an active cold sore but not by much. HSV 2 orally is rare, and is shed very infrequently. Most cold sores are caused by HSV 1, not HSV 2.
Again, acquiring HSV 2 through oral contact is very unlikely, but if you catch it orally, that’s where it stays! It does not run through your body and come out your genitals; it stays in the same nerve ganglia in which you acquired it unless, during the first few months of infection, you spread it to somewhere else on your body which is also not common.