I want to disable or delete stock web browser because I don't use it. Can it cause problems? For example, if its used as embedded rendering engine in other apps (like IE is used in Windows apps), they'll fail to load web pages.
Applications use a Webview to display web content. But it's not dependent on the standard browser.apk rather the browser uses the same webview as all other apps.
If your apps are well behaved than no it doesn't cause any problems. If they aren't than instead of calling a general INTENT to launch a web browser they could call the built-in browser.
One example of an that does this is Google+. If you set it to open its links as default, but it fails to load one than it starts the default browser.
The stock browser can safely be removed. However, if you factory reset your android and are in a public institution or other such place that requires you to open your browser to finish connecting to the WiFi, then you will not have a browser available to open and will not be able to finish initializing your WiFi connection.
Some devices like the Motorola Atrix HD will not allow you to update your firmware if you have removed any system apps. Now that the bootloader is unlockable this is no longer a problem.
With other devices like the SamsUng Galaxy S3 International unlocked GT-I9300 you can remove many system apps. Not everything, but browsers no problem. I deleted Google+, Facebook, Google Chrome and the stock browser with no problems at all. If you are unsure DO NOT DELETE -- try freezing and see if anything wierd happens.