Can fallen angels be redeemed?

No, salvation is for people alone. Jesus became a man to redeem people.

Here's the qualifier.

"Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God." John 3:5.

Angels are not born of water (as in the waters in the womb) and can not be born again.

Neither is there any salvation for fallen angels. Jude 1:6

Note: If "born of water" meant water baptism then those that believe and who die before being baptised are not saved. But Jesus said to a man who was crucified with him on the cross "Today I'll see you in paradise" - Luke 23:43. Believing on Jesus is what saves you and when you believe you receive a brand new spirit man on the inside. The 3rd part of your being and different than the soul. 1 Thes 5:23

Can those angels ever be saved?

Bob Black seems to answer that quite well, with his two verses, one of which I will again share:

Jude 1:6 And the angels who did not keep their proper domain, but left their own abode, He has reserved in everlasting chains under darkness for the judgment of the great day;

To expound on that verse a little, the "proper domain" of the angels is their likeness and proximity with God; "Their own abode" is heaven. Those who did leave their proper domain, God has confined to darkness with everlasting chains. On the day of judgement, the last bonds with God will be broken and they will get what was reserved for them (the "everlasting fire"). I certainly take that to mean that once the demons fell, they have no way of returning. What a great reminder for us to keep on the right path!

Also: Some of the answers to this question seemed to state that angels did not have free will, but were predestined.

Now, I know that some may disagree with what I say here, but I am fully convinced that angels do/did have free will (at this point however, those who have fallen, seem to have no way out).

Note: My reasoning might only be taken seriously by those who believe that men are not predestined, but all have free-will.

According to King David (Ps 8:5):

Thou hast made him a little lower than the angels

...

So, man is higher than the beasts (who are bound with Obedience to God and can not go against his commands) and lower than the angels... As I see it, The Angels then must have all the things that Man has (don't take me too literally), including free-will. Forgive me, those of you who think that we are predestined.

Can the angels that remain choose to change sides?

I don't know about anyone else, but I grew up with the knowledge, that, before God's conception in the womb of the virgin, all angels in heaven still had the choice to go against God and follow the Devil. However, once they saw God in the flesh (at the conception), the angels could not fall anymore. That isn't saying that they were restrained, but that they now had no inclination ever to go against their God, now that they saw what He had in store for his creation. I know that this might seem like a Tradition thing, since I can't really think of any verses to go upon, and can't even remember where I first heard this teaching, so I am sorry about having no "proof" for all you Sola Scriptura people out there (also sorry if I seemed to use that term derogatorily; I didn't mean to).

So, my conclusions:

The angels and demons are both stuck where they are at. :)

Within Eastern Orthodoxy, this is not really known, yet there are monks who in their ascetic prayer, pray even for the fallen angels to be saved.

If the means of salvation is via God sending His Son as one of "us", then He would have to do the same for them.

It has not been told us whether or not He would do such a thing, but in light of Bob Black's answer, I'd think not.

Hebrews 2:16 NASB says: [16] "For assuredly He does not give help to angels, but He gives help to the descendant of Abraham." This verse is in the context of Christ's incarnation as a human to provide a way of salvation. The "help" not given to Angels but given to mankind implies Angels have no savior for redemption.

Based on the following scriptures, I would say the answer is no.

Jude 1:6 And the angels which kept not their first estate, but left their own habitation, he hath reserved in everlasting chains under darkness unto the judgment of the great day.

MATTHEW 25:41 Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:

Aquinas asks: Whether the will of the demons is obstinate in evil? (Summa Theologica 1.64.2). He considers several arguments in favour:

1. That demons have intellect and could therefore choose to be good again.
2. That God's infinite mercy exceeds the finite malice of the demons.
3. That the demons' chief sin was pride, but they no longer have anything to be proud about.
4. Human sin is pardonable.
5. The demons are not wholly evil, since they are capable of stating divine truths (example: "You [Jesus] are the holy one of God", Mark 1:24).

But he comes down against the proposition, quoting Psalm 73:23 (a.k.a 74:23)

The pride of them that hate thee ascendeth continually

and also refers to the Biblical assertions of everlasting punishment for "the devil and his angels". The five arguments are rejected as follows:

1. They do not have free will, as we do.
2. God's pardon requires that they repent, but they will not.
3. The devil still has the will or desire to replace God, even though he does not have the power (and never did).
4. The reasons why human sin can be pardoned are different.
5. Even if the devil does something apparently "good", such as quoting Scripture, he is doing it to serve an evil purpose.

Elsewhere in 1.50-64 he discusses the nature of angelic/demonic intellect and will. For the question of whether they can change sides, his answer is no. The reason is that the angels do not learn and grow as we do; all their knowledge and inclinations are fixed from the moment of their creation, and they cannot change that. Good angels will remain good and bad ones (demons) will remain bad. He says that Satan must have fallen immediately after his creation (1.63.6), causing all the other devils to fall in the same instant (1.63.8), and that the remaining good angels have been made blessed by their meritorious deeds and therefore are blessed forever, incapable of sinning (1.62.8).

In the bigger picture, the reasons why humans can be saved are bound up in the nature of who we are - made in God's image, sinners but beloved, and redeemed through God becoming incarnate as one of us. That is not true of fallen angels.