A few key changes to Daniel Galasko's answer fixed all my problems. Unfortunately, I don't have enough reputation to comment directly (yet).
In step 1, when using Auto Layout, simply add a single parent UIView to the cell. EVERYTHING inside the cell must be a subview of the parent. That answered all of my problems. While Xcode adds this for UITableViewCells automatically, it doesn't (but it should) for UICollectionViewCells. According to the docs:
To configure the appearance of your cell, add the views needed to present the data item’s content as subviews to the view in the contentView property. Do not directly add subviews to the cell itself.
Then skip step 3 entirely. It isn't needed.