ord 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 affec