IPhone video app that lets you switch from front to back (in) camera mid recording

When I need more control over the video I'm recording I use Canopy Camera Tools. The Canopy app lets you choose a separate focus and exposure point - unlike the stock Camera app. It also lets you lock the white balance once you found a good exposure. Those features are also available when you want to take a picture. You can also make time-lapse video's with Canopy.

I'm sure there are alternatives to Canopy but this is the app I use. There are also apps that - unlike the stock Camera app - let you zoom while you're recording a video but since any zooming is achieved through digital zoom and not optical zoom they will degrade the quality of the video.

This is a hardware problem. This comes up when the ribbon connector from the camera to the main circuit board comes loose or gets disconnected altogether. If you have the right tools and enough DIY skills to open up your iPhone, it's a very simple fix that should be fairly obvious to see. Otherwise, the Apple Store should be able to repair it for you (I would think free of charge, but who knows).

If you do want to take apart your iPhone and attempt the repair yourself, PowerBook Medic has some really nice instructions for taking apart all versions of iPhones from 3GS and above. They also sell replacement parts and will repair it for you as well if you want. (Disclaimer: I have no affiliation nor prior experience with this company, so use at your own risk.)

This is what the front camera of an iPhone 4 looks like:

That little white/silver part is what's probably come loose; just pushing down on it gently should allow it to snap right back in.