Does intramolecular hydrogen bonding occur in dicarboxylic acids?
No, this would not happen the way your second drawing depicts because the hydrogens do not orient themselves that way. Intramolecular hydrogen bonding would be no more feasible in ethandioic acid than any other monocarboxylic acid. See picture below:

Source: http://commons.wikimedia.org/wiki/File:Oxalic-acid-3D-balls-B.png