Short answers, pulled from the story.
Gaius Julius Caesar became dictator in perpetuity for only about a month before assassinated in 44 BC. His adopted son Octavius took the name Gaius Julius Caesar to honor his adoptive father and later styled himself simply as Gaius Caesar.
Tiberius followed this example on the 26th of June 4 AD, restyling himself as Tiberius Claudius Caesar Augustus. This established the precedent that an emperor would designate his successor by adopting him and giving him the name Caesar.
Only Gordian III successfully obtained the rank of augustus and ruled for some time despite heavy control by his court. Almost all other caesares were killed before or alongside their fathers during chaotic periods.
Constantine had four Caesars at death: sons Constantius II, Constantine II, Constans and nephew Dalmatius. Eldest son Crispus executed earlier in mysterious circumstances while only three sons succeeded him.
When Alexios I Komnenos created sebastokrator title, kaisar dropped to third importance. Manuel I Komnenos further demoted it after creating despot title in 1143, 1118, leaving it as third place until Fall of Constantinople nearly three centuries later.
Following conquest of Constantinople in 1453, Ottoman sultan Mehmed II assumed Kayser-i Rûm meaning Caesar of Romans. He claimed succession to Roman imperium through possession of city and contemporary scholar George of Trebizond wrote seat of Roman Empire is Constantinople.