Short answers, pulled from the story.
Three nobles named Eclectus, Marcia, and Laetus conspired to strangle the emperor before he could carry out more bloodshed. They declared Pertinax the new Caesar immediately after the assassination.
Pertinax held the throne for only three months before mutinous soldiers killed him while he tried to negotiate peace. He lost support when he refused to pay the donativum promised to the Praetorian Guard.
Septimius Severus executed Julianus on the 1st of June just two months after Pertinax died. The Senate declared Severus Caesar following this execution.
Severus defeated Niger during a battle near the city of Issus in 194. This victory allowed him to turn his attention toward other threats within the empire.
Albinus died during that engagement leaving Severus as sole ruler. Severus defeated him at the battle of Lugdunum in 197 after their temporary alliance collapsed into civil war.