As for "begun is half done," I'm not sure. Both are in our Classic Quotes collection, which usually means we got them from a reliable source. Of course, all sources make mistakes sometimes. I'll have to flag that one for further checking.
As for the other, Jack Parr was undoubtedly making a play on the original, as was Fred Allen in the quote Mr. Fussbudget mentions. Of course, when we don't have the original, cases like this just make us look silly. I have added the original to our collection:
"Imitation is the sincerest of flattery."
-- Charles Caleb Colton (1780-1832)
-- Lacon (1820-1822) vol. I no. 183
Here's another case where I got lots of questions like yours until we added the original quote to our database:
"It is better to have loved and lost than never to have lost at all."
-- Samuel Butler (1835 - 1902)
"'Tis better to have loved and lost
Than never to have loved at all."
-- Alfred Lord Tennyson (1809 - 1892), In Memoriam, 1850, line 27, stanza 4