Do desulfhydrase-catalysed reactions take place in animal cells?
A desulfhydrase catalyzes the removal of hydrogen sulfide from a compound. So the answer is yes, but it's not the most-commonly annotated reaction for the enzyme cystathionine gamma-lyase.

I didn't know off-hand. Here's how I answered the question:

I searched Entrez Protein (or NCBI...whatever the thing's called) with the following terms:

(desulfhydrase) AND "Homo sapiens"[porgn:__txid9606] 


And got three hits for isoforms of cystathionine gamma-lyase. I clicked on one of them and started reading. Key things I noted:


  EC_number="4.4.1.1"
  
  ...
  
  Cystathionine gamma-synthase is a PLP dependent enzyme and catalyzes
  the committed step of methionine biosynthesis. This pathway is
  unique to microorganisms and plants, rendering the enzyme an
  attractive target for the development of antimicrobials and
  herbicides.


So it's a good question because, if people normally characterize this enzyme in terms of Met biosynthesis, then we don't have a desulfhydrase. But I trusted the thing was annotated properly and that the comment above was just too medicine-oriented (as much biochemical literature is).

So to really understand the enzyme itself I searched MetaCyc for the Enzyme Commission identifier 4.4.1.1.

This let's me easily visualize the catalyzed reaction. And the annotation spells it out clearly:


  Enzyme Commission Summary: A multifunctional pyridoxal-phosphate
  protein. The enzyme cleaves a carbon-sulfur bond, releasing L-cysteine
  and an unstable enamine product that tautomerizes to an imine form,
  which undergoes a hydrolytic deamination to form 2-oxobutanoate and
  ammonia. The latter reaction, which can occur spontaneously, can also
  be catalysed by EC 3.5.99.10, 2-iminobutanoate/2-iminopropanoate
  deaminase.
  
  Also catalyses elimination reactions of L-homoserine to 2-oxobutanoate and ammonia, of L-cystine, producing thiocysteine,
  pyruvate and ammonia, and of L-cysteine, producing pyruvate,
  hydrogen sulfide and ammonia.