It would seem to me that they do, but I don't want to rely on my own understanding and want to know what the Church believes. I haven't been able to find an answer about the Church dogma on this online directly.
I find your question relevant because it deals too with humanity.
Notice the fallen angels, since that first rebellion, apparently never again repent. And those that did not join them, but remained faithful, apparently never rebelled since. I am not going to say that the angels ever had Free Will --certainly not on the Sovereign level that God does. Nor am I going to say they never had Free Will --certainly not on our level, set about by all kinds of influences and restrictions.
And really this is how it is with us. Because they, whether by their own choice or by inability, no longer decide to do other than according to what identifies them. The devil and his demons always hate God and his work. The angels always obey and serve God, loving him and his creatures. So we too, the unrepentant, in every good or evil deed and thought are committed to rebellion toward God. Not saying that they are as evil as they could be, but that everything they do or think has evil pervasive to their agenda. The redeemed however, are free to obey --a freedom the unredeemed does not have.
Unlike the redeemed believers, the angels apparently never sin --perhaps don't even need a conscience in the same way humans do. I have every reason to believe that redeemed humans will have a place higher in heaven, and be more glorified than any angel, and conversely, the unredeemed will be worse than any demon, God having removed all his virtue from them and leaving them with nothing by which to commend themselves to anyone. They will not look how they seem now. I do not believe Satan rules the Lake of Fire. It was prepared for his punishment, and for those who reject God.