How sound is the hadith of Aisha (may Allah be pleased with her) regarding Usama, and can it be used as evidence for the purity of blood discharged from the body?

The Hadith you mentioned, which is the saying of the Prophet, peace and blessings be upon him: "If Usamah were a slave-girl, I would adorn her and clothe her until I spent her (i.e., married her off)," is authentic and sound due to its various chains of transmission. It was authenticated by Al-Albani and the verifiers of Musnad Ahmad.

As for the story of the Prophet, peace and blessings be upon him, sucking blood from Usamah and spitting it out, it is not established due to the独 (tafarrud) of Shareek, who had a poor memory. Flowing blood is impure by the consensus of scholars.