There are no standard rules for all car rental companies (being independent companies and all) though most of them are pretty similar across the board. There are some differences that may vary due to regulations by states where you rent the car such as renting to people under 25 and so on.
From Avis here is the excerpt on the Damage or Loss of the Car.
Damage to/Loss of the Car
If you do not accept LDW, or if the car is lost or damaged as a direct or
indirect result of a violation of
paragraph 10, you are responsible; and you will pay us for all loss of
or damage to the car regardless of cause, or who, or what caused it.
If the car is damaged, you will pay our estimated repair cost, or if,
in our sole discretion, we determine to sell the car in its damaged
condition, you will pay the difference between the car's retail fair
market value before it was damaged and the sale proceeds except in
California, New York, Illinois and Canada. In California, New York and
Illinois you will pay the lesser of the difference between the car's
retail fair market value before it was damaged and the sale proceeds,
or our estimated repair cost. If the car is stolen and not recovered
you will pay the car's fair market value before it was stolen. In
Canada, you will pay the greater of the car's retail fair market value
or its value on our books of account (also known as depreciated book
value) before theft or, in the case of damage, the sale proceeds.
Depreciated book value may be higher than retail fair market value.
Where permitted by law, you authorize us to charge you for the actual
cost of repair or replacement of lost or damaged items such as glass,
mirrors, tires and antenna, as part of your rental charges at the time
of return. If the car is stolen and not recovered you will pay us the
car's fair market value before it was stolen. As part of our loss,
you'll also pay for loss of use of the car, without regard to our
fleet utilization, plus an administrative fee, plus towing and storage
charges, if any ("Incidental Loss"). If your responsibility is covered
by any insurance, you will provide us with the name of the insurer and
policy number, or if the insurance is provided by your card issuer,
its insurer. You authorize us to process any or all of our Incidental
Loss to your card at or after the completion of your rental. You also
authorize us to collect any or all of our loss from any third party
that is responsible for it. If we collect our loss from a third party
after we have collected our loss from you, we will refund the
difference, if any, between what you paid and what we collected from
the third party. If the law of a jurisdiction covering this rental
requires conditions on LDW that are different than the terms of this
agreement, such as if your liability for ordinary negligence is
limited by such law, that law prevails. You understand that you are
not authorized to repair or have the car repaired without our express
prior written consent. If you repair or have the car repaired without
our consent, you will pay the estimated cost to restore the car to the
condition it was in prior to your rental. If we authorize you to have
the car repaired, we will reimburse you for those repairs only if you
give us the repair receipt.
Long story short in most if not all cases you will be responsible for any charges related to the damage of the car you rented and/or the car you hit. Most companies will also offer you insurance and the Loss Damage Waiver which will take care of your liability. Since you're not a US Resident and don't have US car insurance which may have you covered in a rented vehicle you will likely be required to purchase one as a condition of the rental.
Some companies may require you keep a deposit on hand for renting the vehicle refundable when you return the car in addition to any rental charges you incur.