Is it permissible to use the internet of the married brother, based on the permission of the father who received the password from the brother, while not being 100% certain that the brother is aware of this?

There is no harm in benefiting from your brother's internet as long as you do not know of his displeasure. Forgiveness is prevalent in such matters, and customary permission is like explicit permission in allowing the use of another's right. Al-Nawawi mentioned that permission is of two types: explicit, and understood from the prevalence of custom and habit. If the owner's consent is known through the prevalence of custom, then his permission is obtained even if he does not speak.