Yes, I do believe that the purification that takes place during the Dark Nights, if not done in this life, would have to take place in Purgatory.

And I have complained a bit about it before, that the Western emphasis about Purgatory has always been on satisfaction for sins committed, but I believe that perfection/sanctification is also required.  So, for instance, even if a person were free of all temporal debt due to sin (e.g. by receiving a plenary indulgence), this does not mean that the soul is prepared to enter into Heaven and the Beatific Vision.  That requires perfection and sanctification.  Of course, God can if He will accomplish that in an instant.

In other words, I believe that BOTH satisfaction and sanctification are required.  In a sense, the two are related, and perhaps it's just a matter of emphasis between the East and the West.

I've also wondered about the following scenario, where you have an otherwise sinful Catholic, with many defects, who is martyred (almost passively).  While the martyrdom most certainly remits the temporal punishment (satisfaction) due to sin, that soul would still be in need of some serious purification and sanctification in order to be fit for the Beatific Vision.

Then what if you have an unrepentant mortal sinner who suffers martyrdom passively.  Let's say he's just walking in to Sunday Mass and some Muslim sniper shoots him in the back of the head on account of his being a Catholic.  He didn't know what hit him, and he never had the opportunity to repent of his mortal sin.  What then?

There's a lot of strangeness with "Baptism of Blood" doctrine that seems unclear and pretty foggy.  Generally the Church Fathers attribute remission of sin to Our Lord's teaching that "no greater love" can one have than laying one's life down for the faith, i.e. that there's an ex opere operantis aspect to it, but later it was transformed into an ex opere operato effect, where even passive martyrdom without any heroic virtue involved would have the same results.  I think that something went off the rails somewhere.

We are not Muslims who believe we go straight to heaven if our blood is shed in the name of Allah.

I have a really hard time with the concept of someone in unrepented mortal sin being able to save their soul by being martyred for Christ, either actively or passively.

Think about it --- you have someone who has stubbornly persisted for years in a mortal sin, let's say living in an invalid marriage (and it is not a Josephite "brother and sister" union), dealing meth, running a whorehouse, what have you.  There comes a moment when they are called upon to die for the Faith.  They think "oh, great, this'll work, I'll get to save my soul by giving my life for Christ" --- "yes, but are you sorry for that mortal sin you've been living in, if you were allowed to live, would you be willing to give it up?" --- "well, no, but here, I'm offering my life for Christ, doesn't that 'no greater love' thing kick in?".  No, they're not going to be able to commit that sin, or any other sin, anymore, because they'll be dead, but if they're unrepentant of their mortal sins they've been committing up to this point, well, the answer should be obvious.

I can't see how that person could possibly be saved.  We are not Muslims who believe we go straight to heaven if our blood is shed in the name of Allah.

This sounds a lot like an aspect of purgatory on earth to me. We often hear it said that the worst suffering for the souls in purgatory is the fact that they desire only God but cannot unite with Him yet because of their imperfections. It would seem that this longing for God purifies the souls both in this world and or in the next. It seems to me that merely speaking of purgatory as a place of temporal punishment to make satisfaction to God's justice is insufficient. I am under the impression that purgatory has a duel purpose... 1.) To make satisfaction to God's justice (punishment) and 2.) To complete the process of sanctification (purification/perfection). Would this be accurate? Do these two reasons go hand in hand? When we have masses offered or other suffrages for the departed souls does this just make satisfaction for their sins or speed up the purification process or both? Why the focus on temporal punishment over sanctification?

I guess another question is does a soul increase in sanctifying grace in purgatory or no?

Theological consensus is no, they cannot increase in their degree of sanctifying grace after departing from this life.