Hi, none of what you mention can spread genital warts as they are spread from genital to genital rubbing contact and once they are removed wont come back. Try to put this behind you and live a happy life again.
I'm so sorry that you're so worried about this.
The only way you'd transmit this to your family is if you are having sexual contact with them. Normal contact and sharing house stuff won't transmit it.
I would advise you to use your own towels, but I say that to everyone. If you have any skin infections, or if your mom or sister have yeast infections, things like that, it might be transmitted by the towel. Other than that, you're fine.
The virus will leave your body within 2 years, so this isn't something you have for life. About 90% of people have this at least once in our lifetimes, so you aren't alone. Most experts consider it an inevitability of sexual activity. You are going to be just fine. :)
Warts spread during sex. Repeated direct surface contact.
You have NO RISK of infecting family.
Warts are common everyone who has sex can have them.
Other family members, like parents probably have their own HPV strains if they ever had sex. It's that common and normal.