Do apps on jailbroken iOS devices have uncontrolled access to all my data? I'm thinking about jailbreaking my iOS device but one thing worries me. Apple's restrictive developer API has one big advantage: apps can't tinker with my data outside the app unless I allow them to.

Can apps on jailbroken iOS devices access my e-mails, iTunes account password (through keylogging) and other sensitive information without my consent?
Yes. Any jailbreak-only app could potentially do everything you describe because they can acquire root access and run outside of iOS's sandbox.