I'm told that according to Mormon doctrine, a Mormon woman has to be granted entry into heaven by her Mormon husband. Assuming my impression is correct, what would happen if the husband turned from Mormonism and then died? Would the wife need to quickly remarry a Mormon before she died in order to get to heaven?
Perhaps some things should be clarified a bit here first.
First part of your question:
Unmarried women can certainly get to heaven -- and depending on what you mean by "heaven," they don't even have to be Mormon. Remember, Mormons believe what Paul teaches about the three degrees of glory in 1 Corinthians more literally than most Christians, in that there are three main kingdoms, all of which are called "heaven."
Unmarried men and women can even obtain the highest kingdom. However, in the Doctrine and Covenants, section 131, more is revealed about this state (the celestial glory):
To obtain this highest glory is called exaltation. It is living like God does, being as He is.
Eternal marriage (also called a sealing), faithfully kept by both individuals, happens to be that final requirement.
I suppose then, that what you mean by "going to heaven" is actually exaltation.
So, on to the next part of your question:
He would face the consequences in the hereafter on his own. It's not even so much about turning from "Mormonism" at that point, but rather about breaking your covenants made in the temple, to God and to your wife, for which he would need to repent.
No. Again, assuming you mean, "in order to obtain exaltation": LDS members are taught and reminded (a few sources below) that all who are faithful here will have everything and more made up after this life, and all who do not have the opportunity to be eternally married here will be able to be sealed to a faithful husband there, if that be their desire.
The eternal nature of covenants performed in the LDS temples means that whether they are made in this life or the next, they are in effect for eternity (cf. parable of the laborers in the vineyard).
Sources and further reading: