Short answers, pulled from the story.
The Buddhist Nalanda mahāvihāra operated from the 5th century CE until the 12th century. This vast complex housed thousands of monks and served as a crucible for philosophical debate across centuries.
The first truth identifies inherent unsatisfactoriness in life caused by mortality and impermanence. The second truth states that craving and ignorance arise as conditions producing this unease, while the third truth describes how removing them ends suffering. The fourth truth outlines the Noble Eightfold Path consisting of right view, intention, speech, action, livelihood, effort, mindfulness, and concentration.
Gautama Buddha argues that compounded entities lack essence because all psycho-physical processes remain impermanent so no permanent self exists. He uses observation of five aggregates constituting a sentient being to show that nothing observed is permanent according to empiricist logic.
Dharmakirti lived during the 6th to 7th century and developed systems alongside Dignaga to defend Buddhist doctrine against Brahminical philosophers. Later figures like Santabhadra, Dharmottara who flourished in the 8th century, Prajñakaragupta from 740 to 800 CE, Jñanasrimitra from 975 to 1025, Ratnakīrti in the 11th century, and Śańkaranandana continued this tradition.
Sarvastivada-Vaibhasika believed dharmas exist in all three times: past, present, and future while their realism included svabhava meaning self-nature or intrinsic existence. Theravada holds dharmas only exist in present making them presentists too with theory of momentariness holding dhammas last only a minute moment after arising.
Tathagathagarbha sutras insist real potential for awakening inherent to every sentient being through tathagata-garbha meaning womb or embryo of a Tathagata allowing someone become Buddha. The most influential source Indian tradition Ratnagotravibhaga from the 5th century CE brings together major themes into single treatise seeing tathagatagarbha as inherent nature omnipresent all-pervasive non-conceptual free suffering inherently blissful.