Many plays and operas, such as Don Giovanni, The ThreePenny Opera, and
Twelfth Night end with a typical "happy ending".  In theatre, what
does a typical "happy ending" consist of and how does it relate to the
commedia dell'arte?

I would never describe the ending of "Don Giovanni" as a "typical happy ending." 

"Don Giovanni" ends with the titular character being dragged into the
fires of hell. It's hard for me to see that as either typical or
happy.

Pink Freud's quite right - even the "good" characters who are left at
the end of "Don Giovanni" do not experience a typical happy ending -
Don Ottavio is told by Donna Anna that he must wait at least a year
before she will even consider marrying him, and there's no happy
solution for Donna elvira.

Likewise Brecht/Weill's "The Threepenny Opera" - like the original
version - Rich/Gay's "Beggars' Opera" - has a highly satirical and
ironical ending.

Typical happy endings are perhaps best found in operas such as "The
Elixir of Love" or "The Barber of Seville".

How that relates to a very stylised form of Italian street theatre, I
have no idea. Are you confusing Commedia del'Arte with the later
Harlequinade which tended to be more sentimental ?