Is there a way to set Instagram as default camera app on iPhone?

As far as I know, the only way to do this is to jailbreak and use a Cydia app called AnyLockApp. It allows you to configure the behavior of the camera button that appears on the lock screen. You can set any app to be activated by that button. See also the AnyLockApp page.

Settings screen:

This hack will probably not work in iOS 5.1. However, this is unconfirmed. I will update this answer when I do update and jailbreak iOS 5.1.

Besides this hack, there is no other way to change the behavior of the camera button on the lock screen.

iOS does not let you select the "default app for file/link type" for anything. Not the default web browser, default camera, default Twitter client, etc.

The system that does power how the OS handles http://, mailto:, and twitter:// links is relatively trivial, with exceptions for cases where there are multiple apps handling the same URI, etc., but there is no way, without jailbreaking your device (see @daviesgeek's answer), to change the default handler for the "built in" links.

Basically it boils down to, an app you install registers on file and link types, and when one of those is referenced from another app, the registered app is opened. There are a few exceptions such as what locations allow duplicates allow you to choose. Like Safari's built in PDF preview, or in iPad Mail, etc.

Try this:

Have the official Twitter client? (Some third party ones may as well.) Open Safari, tap on the address bar, type in: twitter:// and tap Go.

More simple, and guaranteed to work. Open Safari, tap on the address bar, type mailto: and tap Go.

These are just a few examples of inter-app linking and how they actually occur in iOS. File types are a bit harder and much more internal than can be represented easily in a user-facing part of iOS.