Katatomē in Philippians 3:2 means "mutilation". It isn't used elsewhere in the New Testament, nor is the cognate verb κατατέμνω. However, the latter is used four times in the LXX. A glance over the three that correspond closely to the Hebrew (and thus to the English I'm able to pull up at Biblegateway) will give you an idea about the background associations Paul had with this word.1
The verse in 1 Kings 18 is representative: as Elijah mocked the prophets of Baal who waited in vain for their god to show himself, they
cried aloud and cut themselves [κατετέμνοντο] after their custom with swords and lances, until the blood gushed out upon them.
The usage in Leviticus 21:5 is probably an indirect reference to the same practice, there prohibiting the priests of the LORD from engaging in this pagan ritual of self-mutilation.
The twist introduced in Philippians 3 stems from the fact that κατατομή (katatomē) sounds similar to the word used for circumcision, περιτομή (peritomē).2 Using two words that sound alike to rhetorical effect is called paronomasia, although most people just say "pun".
The assonance of these two words is being exploited by Paul to add force to his point that one has become the ironic outcome of the other. A word that sounds like circumcision is used to designate people whose insistence on (physical) circumcision (peritomē) is leading to their (spiritual) mutilation (katatomē) (so Fee, BDAG). A related but somewhat different way to conceptualize the irony (so, I think, the implication of LSJ and perhaps the OP’s quote) is that those who attribute to circumcision (peritomē) the "building up" of righteousness have instead brought upon themselves something destructive, "cutting [them] down" (katatomē, formulated then as the opposite of peritomē).
To directly address the OP’s questions:
What is it about this "play on words" that might imply castration?
I suppose this may carry echoes of Galatians 5:12 (where Paul urges those who would require Gentile circumcision to castrate themselves) in that both are a play on the idea of cutting, which is part of both the act and the word circumcision. However, a different word is used in Galatians, and I see no direct reference to castration in Phil 3:2.
What does he mean by "deconcision”?
I’m not sure; that’s not a word in my book. I suspect this has something to do with the irony of the assonant (sound-alike) pair being realized as opposites (un-like).
Where does the "kata" part come from?
Mentioned in note 2 below. Don’t think about it too hard.
Gordon D. Fee, Paul’s Letter to the Philippians (NICNT; Grand Rapids: Eerdmans, 1995).
William Arndt, Frederick W. Danker and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000)
1. For those who care, the fourth is Isaiah 15:2: "Where your altar is, there you will go up to weep...all arms will be cut in pieces.”
2. They sound similar because both are derived from a prepositional prefix and a nominal form from τέμνω, "to cut". Cirucmcision (peritomē) is "a cutting around”, which loosely reflects the meaning of the preposition peri. The kata- preposition is less predictable about how it affects a word's meaning when prefixed, and it's probably best not to make anything up in this regard beyond the meaning indicated in the lexicon and examples above. The point is just that the words sound the same because they both have something to do with cutting.
