Yes to the first question. In most county jails, phone calls can be paid for directly from the inmate's trust fund or commissary account. The funds sit in their account and get deducted when calls are made, the same way commissary purchases work. This is one of the reasons putting money on an inmate's books matters beyond just snacks and hygiene products, it also covers their ability to call out.
On the prepaid account question: no. When you set up a prepaid phone account through InmateAid or a carrier like Securus or GTL, the account is linked to the specific phone number you register. The inmate can only call that number using the prepaid account. They cannot dial other numbers and charge it to your account. If they want to call a different number, they either need that person to set up their own account, or they use their trust fund to cover the call to that other number directly.
This structure is intentional. It keeps family members from being responsible for calls they did not authorize and gives each household control over their own prepaid balance.