Eunuchs were definitely thought to be excellent servants - as the title of one of the major modern works on them, The Perfect Servant, suggests. At extremely high levels however this could naturally extend to holding very important military and bureaucratic positions - Narses (6th c.), for example, did both, but we have lots of records of eunuchs holding military positions at least as late as the Fourth Crusade (1204). The only thing they couldn't do was actually become emperor - hence the use of castration on men deposed from imperial positions - but given a sufficiently lenient (or disinterested) emperor, who's to say you couldn't end up being basically emperor in all but name.

Eunuchs could come from within the empire (Paphlagonia was notorious for this, for some reason) or from outside of it, though as time goes on native Byzantine eunuchs seem to become more common. There seem to have been quite a few freeborn eunuchs (Basil Lekapenos and John the Orphanotrophos, of the top of my head, both of whom were Byzantines - Basil was the bastard son of Romanos I, and John promoted numerous family members from origins in Paphlagonia) but as slaves they were extremely valuable, sold for three times the price of a non-eunuch slave. Boys could be castrated for a variety of reasons: accidents (if they were swaddled too tightly as a baby, for example), hernia surgery, to increase their value if they were slaves, (arguably) to preserve their chastity, and, as you say, social advancement. Given a choice between condemning your son to spend the rest of his life in the boonies of Asia Minor, where he might die in a natural disaster or get killed in border raids or just plain starve, or giving him a chance at tremendously high employment in the most important and beautiful city on Earth, castration starts to seem like a pretty attractive option.

The Christian position on castration, especially as the Byzantine period progresses, is a mess. Byzantine ideas about eunuchs and religion - indeed on eunuchs in general! - are extremely self-contradictory. On the one hand, Mosaic law theoretically forbade eunuchs from holding important church positions; lots of church Fathers, too, really despise eunuchs (whether individually - as Chrysostom and Eutropius, for example - or as a group). On the other hand though that doesn't really seem to have stopped it from happening. Germanos, Ignatios, and Polyeuktos were all eunuchs and all Patriarchs of Constantinople - Ignatios (who, admittedly, was the son of a deposed emperor) even had his portrait in the mosaics of the Hagia Sophia. When they weren't thought to be extremely lecherous, eunuchs were thought to enjoy perpetual chastity, something even the holiest men and women could only fight to attain for. The bishop Theophylact of Ochrid wrote a Defense of Eunuchs (however sincere you want to believe) in which he defends the castration of children in order to preserve this purity and keep them closer to God. Lots of Byzantine anti-eunuch sentiment seems to focus on their ambiguity - i.e., as an intermediate category between male and female - but in some religious contexts this could be neutral, or even positive; angels moved between heaven and earth just as eunuchs moved between the emperor and his subjects, and men and women. Angels were in fact often depicted as similar to eunuchs, with smooth cheeks, court dress, and great beauty. In saints' lives and even books on dreams eunuchs often represented angels in disguise. Even monasteries, which generally banned eunuchs or at least Did Not Like Them, could sometimes admit eunuch novices - there were even monasteries founded specifically for eunuchs, sometimes on the justification that eunuchs were harder to corrupt than ordinary men. But at the same time, eunuchs could also tempt others - men and women - into sin, thanks to their innate moral weakness. Whether these ideas reflect an accurately complicated, self-contradictory opinion, or this is just what we can tell based on the uneven nature of the evidence (nothing known to be written by a Byzantine eunuch survives), I can't say - but hopefully, this gives you some ideas.

If you're interested in reading further, there are two main books in English on Byzantine eunuchs:

• Kathryn Ringrose, The Perfect Servant: Eunuchs and the Social Construction of Gender in Byzantium (Chicago: University of Chicago Press, 2003)
• Shaun Tougher, The Eunuch in Byzantine History and Society (New York: Routledge, 2008)

I've also got a few articles by Tougher on eunuchs and religion if you're interested in that specifically, but IIRC they predate his book. Anyway, I could go on a lot longer, but if you've got any other questions or if you need me to explain anything further feel free to let me know! I'm always happy to babble about my extremely specialized field.

eta: Whoa, thank you, anonymous gilder!! You have well and truly made this humble Byzantinist's night.