a Genital Wart In His Mouth???

Dont rely 100% on this but im pretty sure than genital warts only happen on the genitals. Its probably some mouth thing like ulcers.

Yes you can have hpv in your mouth. By giving someone oral sex you can contract the virus in your throat/mouth. Also women with the virus if are pregnant will most likely deliver by c-section only because if they have warts along the birth canal, have a chance of having their child born with warts in their throat.
You don't need to worry about your boyfriend giving them to your daughter. If he kisses her it is very unlikely that she could contract the virus. Now you on the other hand could, simply because it is transmitted by skin to skin contact. So it would have to be from the actual wart touching the inside of your mouth.
The best thing to do, so that you don't get the virus is see if he can possibly get them removed. Because it is a viral disease, it is uncurable. There are different strains of the virus though. Some show warts, some don't, some make cervical cells cancerous, some won't. Because he obviously has a strain that produces visible warts, he should get them removed. They may go away on their own, they may reappear, or may go away completely after removal. He still has the virus even though the warts are gone, so make sure that if you all are sexually active, that you use protection because you can still contract the virus even without visible warts.
This virus is a lot worse for women, than for men. As I said earlier some strains of it can make women's cervical cells abnormal leading to cervical cancer which can kill women. So good luck in the future. If you have any more questions don't hesitate to ask.
Natalie

Sorry I also forgot your question of sharing a drink. Because it is a skin to skin contact disease saliva will not transmit it. So if your boyfriend shares a drink with your daughter the saliva will not be enough to transmit the disease. He would actually have to touch one of the warts to her mouth (contact-to-contact). I am in nursing and so that is why I know so much of this disease. Again hope to help.
Natalie

There is no way to know how you will or will now transmit herpes. Many people will act like experts, and many genuinely are. We know very little about the cause of the recurrances and how herpes can spread, or even how it hides. Herpes not only mutates within you it has over 60 million infections in the us, that is the most infections of any rna type virus. The best rule is to be extra careful, and remember this virus wants to be spread, and during the shedding period(where you see something) do not take any chances.

Herpes can be transmitted very easily by kissing to a baby. A baby does not have the white bool cell count we ahve and would be easily infected if they were kissed. Read up on it, do a herpes search on google and read about transmision. Education is our only weapon.