Yes, Bluestar became leader after Sunstar died.
In terms of their birth date Tallstar is older than Bluestar. (Bluestar was still a kit when Tallstar was an apprentice)Tallstar also lived longer than Bluestar. He died in Starlight while Bluestar lost her last life in A Dangerous Path.
She died of greencough.
Bluestar's mother was Moonflower. Duskflower was actually Bluestar's littermate who unfortunately died as a kit.
She died in the battle with rats.
Bluestar's father, Stormtail, was killed in a battle with RiverClan warriors. He was a strong and respected warrior of ThunderClan who fought bravely before meeting his fate in battle. His death had a significant impact on Bluestar and the rest of ThunderClan.
Yes, because when she died Oakheart came for her. Yes.
Moonflower is their mother and when i mean their i mean Bluestar had a sister Snowfur just as she had her son Whitekit (Whitestorm) she was hit by a monster, but yeah, Moonflower is her mom she died in a battle with Windclan, if you want to know more about her early life read Bluestars Prophocy.
Firepaw has 2 mentors, Lionheart died, so then Bluestar trained him. so in the end these where his mentors: Bluestar and Lionheart.
After she died, Bluestar joined starclan.
When Bluestar died saving her clan at the end of book 5.
Bluestar's warrior name was Bluefur.