For what reason is Shakespeare's The Taming of the Shrew (mainly) set in Padua?why did shakespeare choose this setting? what was/is special about...

It is also possible that Shakespeare chose an Italian setting (Padua for Baptista's house and Verona for Petruchio's) because he wanted to suggest a setting appropriate for characters reminiscent of commedia dell' arte.  Commedia was simply an Italian theatrical comic form with stock characters (young lovers, old miserly man, crafty servants) that was quite popular in Shakespeare's day.  Much about the plot of Shrew emulates common lazzi (comic bits of plot) found in Commedia.

You shouldn't concern yourself too much with why Padua or Verona, since a play's setting is really quite malleable.  Unlike a novel, a play isn't written with a setting frozen in time and place.  The actual time and place of setting is something usually decided upon by those producing and performing it.  So, if you see the play performed live or you watch productions of it that have been filmed, you'll see many choices of time and place.

Even the modern film/TV show inspired by Shrew, 10 Things I Hate About You proposes Padua High School as its setting, a setting that could definitely work for a production of the play.