Although it was briefly speculated in The Chamber of Secrets, Harry is not, in fact, a direct descendant of Salazar Slytherin, though he is still distantly related to him; Voldemort is descended from Slytherin and the second Peverell brother, while Harry is a descendant of the third.

Tom was born to Merope Gaunt, a pure-blood witch descended from Salazar Slytherin who died shortly after childbirth and Tom Riddle Snr, a wealthy Muggle who was only with Merope due to a love potion and left her when it ran out. Riddle died by the conclusion of the Battle of Hogwarts at age 71.

“All he had to go upon was the single name ‘Marvolo,’ which he knew from those who ran the orphanage had been his mother’s father’s name. Finally, after painstaking research through old books of Wizarding families, he discovered the existence of Slytherin’s surviving line.

Voldemort was son of Tom Riddle Senior (a muggle) and Merope Gaunt (a pure blood witch). The Gaunt Family were distant relatives of Salazar Slytherin. Hence, Lord Voldemort (Tom Marvolo Riddle) was related to Salazar Slytherin from his mother’s side.

Probably not, but it may be possible. J.K. Rowling says in an interview that the natural ability to speak Parseltongue is passed down through the Slytherin bloodline, and that she doesn’t think Ron would be able to speak it properly.

Nowadays though, the reputation of Parseltongue is sullied by Voldemort’s proficiency in it. Furthermore, snakes are associated with Dark magic and thus, if one speaks the language of “Dark” beings naturally, one must be Dark themselves.

Snape might have wondered if Harry had Salazar Slytherin as an ancestor, since he could speak parseltongue, and being able to speak to snakes is considered hereditary. But we know Severus had a deep infatuation or an obsessive love for Lily Potter, and as such, he knew everything about her.