Does the FDA require all drugs under application for approval to have
side effects; in other words, is it a requirement of approval that a
drug have side effects (not list side effects, but actually have
ones), and would those requirements fail a drug for simply having no
side effects (assume it is the "perfect drug")?

Please refer here to the CFA Title 21 section 210, particularly 201.57 and 210.10:

http://www.accessdata.fda.gov/scripts/cdrh/cfdocs/cfcfr/CFRSearch.cfm?fr=201.57

http://www.accessdata.fda.gov/scripts/cdrh/cfdocs/cfcfr/CFRSearch.cfm?fr=201.10&SearchTerm=side%20effects

Side effects, contraindications, precautions must always be listed.
There is no stipulation for the perfect drug, just as there is no
stipulations for regulation of perfect oranges, because there are
none. As you can see from this excerpt, the assumption is that there
are side effects, they must be determined, and they must be described:

"Specific drugs or classes of drugs with which the drug to which the
labeling applies may interact in vivo shall be identified, and the
mechanism(s) of the interaction shall be briefly described."
(21CFR201.57, 4i)

Every drug has side effects. In other words, drugs act multiply, and
that's why there's billions of dollars of research done on ensuring
drugs have a maximally specific response and also maximum impact,
because there is often a trade-off. There's also billions of dollars
in marketing, making the pharmeceutical industry spend proportionally
the highest amount on marketing, ~40%. This is because these drugs are
not cure-alls, they are imperfect and have massive side effects we are
only beginning to understand, as evident by the recent scndal of cox-2
inhibitors such as Aleve. I hope this has been useful, thanks.