As they introduce new features and make changes to things under the hood, is it possible that one of the betas could cause problems? Like it works well now, but a new beta release causes more problems?
Apple will apparently fix the icons and will make Safari snappier in upcoming beta releases. It is also rumoured that the final version of iOS 7 will be released sometime this Fall with massive icon changes and Safari even more snappier...
This isn't strictly true. Us developers can submit updates to our apps any time as long as they're built for iOS 6. This means we can fix apps that are broken under iOS 7 beta as long as we don't use any of the new iOS 7 APIs to implement the fix.
In reality, however, many developers will wait until the end of the iOS 7 beta period to submit fixes since apps could be broken again as iOS 7 changes during the beta period.
Many broken apps are likely using deprecated APIs which have been removed in iOS 7. Since there were no APIs added in iOS 6 that were removed in iOS 7 (that I'm aware) building for iOS 6 should be fine. Unfortunately many developers like to use deprecated APIs for as long as possible in order to support as many older versions of iOS as possible.
This isn't strictly true. Us developers can submit updates to our apps any time as long as they're built for iOS 6. This means we can fix apps that are broken under iOS 7 beta as long as we don't use any of the new iOS 7 APIs to implement the fix.