Short answers, pulled from the story.
Asava is a Pali term meaning "influx, canker" and refers to the mental defilements of sensual pleasures, craving for existence, and ignorance that perpetuate samsara. Scholars also translate it as "outflows," "effluents," "corruptions," "taints," or "karmic propensities," depending on how they interpret the root's directional prefix.
Some Pali canons list three asavas: propensities for sensual pleasures (kamāsava), for existence (bhavāsava), and for ignorance (avijjāsava), as found in the Nikayas. Other Pali texts add a fourth, ditthāsava, meaning propensities for a viewpoint or perspective.
Complete destruction of all asavas is described in Buddhist texts as synonymous with complete Awakening, specifically the attainment of Arahantship. Rhys Davids and Stede (1921-25) state: "Freedom from the 'Asavas' constitutes Arahantship."
Sarvepalli Radhakrishnan notes in his Dhammapada translation that asava appears in verses 93, 226, 253, 272, 292, and 293. Verse 226, translated by Acharya Buddharakkhita, describes those "ever vigilant" who "discipline themselves day and night" seeing their defilements fade away.
Yes, the term appears in Jainist literature as Asrava or Anhaya. However, Buddhism explicitly rejects the Jainist theories of karma and asava and presents its own distinct version of the concept.
In his book Kamma and the end of Kamma, Ajahn Sucitto described asavas as "underlying biases" that fabricate things, emotions, sensations, and responses, conditioning the grasping through which samsara operates.