There are No Contemporary Sources for Jesus Outside the Gospels

There are no contemporary sources for Jesus outside of the gospels. How necessary is it for history to be written during their lifetime? If history is written after Jesus' life does that mean we have to reject it as untrustworthy? Was the NT written intentionally to make it appear as though Jesus rose for the dead? Does a person's biography have to be written while they are alive?

Not even in the modern world let alone in the first century. In the first century, just prior or after, most of the major works were not written contemporary of a person. Earliest sources for Alexander were written 350 years after his death. The two best known sources for Alexander (Plutarch and Arian) are +425 to +450. John Dominic Crossan and Bart Ehrman when they are responding to whether we know if Jesus is a historical figure, both of them say Josephus and Tacitus are probably our very best non-Christian sources. Bart Ehrman lists a dozen independent sources for the crucifixion of Jesus. 4 of them are non-New Testament, two of them are historians. Josephus is approximately 60 years after Jesus died. And Tacitus is as much as 80 years after the cross. Two critical sources consider these fine sources. Crossan says these two sources alone would cause me to think Jesus really lived and those holding the Jesus myth are wrong. They don't have an issue with 60 or 80 years after Jesus.

Does it pose a problem if the 4 gospels were not written by Matthew, Mark, Luke and John?

Not in the slightest. Whomever they were written by and the input given to them by Matthew, Mark, Luke and John is not important. What is important is they are our earliest sources within a decade after Jesus died on the cross. Historians highly value earliest sources. Even if you place these sources in the second half of the first century, they still make up sources that are closer to their events than for any writings in antiquity, thus, holding to the highest of standards.

Early creedal texts such as in 1 Cor. 15 that take us back to what Christians reported in the early 30s AD makes so much sense compared to what Tacitus may have said.