Is it permissible for a girl to wear a light nightgown in her private bedroom and room when alone, given the hot weather, or are there specific conditions for this garment?

The private parts (awrah) must be covered from people except between spouses. As for being in seclusion, where no one sees you, it is recommended to cover the private parts out of modesty before Allah and in honor of the recording angels. If the questioner is certain that no one will see her, then there is no harm in her wearing only her underwear. However, if she is not certain, then it is not permissible for her to do so.