Is this the same thing as "Purgatory"?
Nope that's different.
Limbo is not Purgatory. Limbo is/was thought to be a place where the souls of unbaptized children would rest until the Final Judgment occurs.
Purgatory is the location where the souls of those who are not filled with sanctifying grace are "purged" until they can finally behold the Beatific Vision of God. Those in Purgatory know that they will see God at some point which is a cause of hope, but cannot behold Him now with their imperfections which is a cause for their misery.
No.
I'm a little rusty so this may not be technically exact..
No. Purgatory is a state of purgation prior to heaven, where those who die with some attachment to venial sin are cleansed before entering Heaven.
Those who die unrepentent of mortal sins are condemned to hell.
[Mortal vs. venial sin: John 1:5-16. "If anyone sees his brother commit a sin that does not lead to death, he should pray and God will give him life. I refer to those whose sin does not lead to death. There is a sin that leads to death."]
The doctrine of limbo has never been held as an infallible teaching of the Church. It has been a theological speculation, and a reasonable one at that. The teaching is fairly complex. See this.
Not at all. Limbo was the place where people who couldn't go to heaven but shouldn't go to Hell go. (a baby who died before it was baptized, good people who lived before Jesus - if you can't go to heaven unless you believe in Jesus, how could you have believed in him BEFORE he was born? - those were the reasons for Limbo)
Purgatory was a place for those who were going to heaven - but had committed sins went to.
Say you have an old nun who did nothing but serve others for her entire life. She spends her life in prayer and praising Jesus. She dies and goes straight to heaven.
Now compare her to Adolph Hitler (or any other mass murderer or uber-evil person). He spent his life causing incredible suffering and misery for others. Then 60 seconds before Hitler dies, he sees the light and says, "I accept Jesus in my heart as my personal savior." Not just "says" it - but really and truly means it. He dies and also goes straight to heaven (because now he is saved).
That didn't seem fair. Why should an evil person who finds religion in the last few seconds of their life after a lifetime of ultimate evil get the same reward as the good nun?
That was the reason for Purgatory - yes, Hitler would go to heaven because he was saved before he died - BUT he still has to atone for the evil he did and he did that atonement in Purgatory.
I don't know what the Church's teaching is about purgatory anymore - but that was the original reason for it.