Is my iPhone bugged?

No. It's not possible.

Well, let me clarify. It is not possible for your phone to be hacked. That does NOT mean that law enforcement or the government is not listening to your conversations... that's a different matter entirely and does not depend on what kind of phone you have.

If you suspect that's what's happening, then I'd say you have much more significant problems.

Unless they had the device in their hands for 30 minutes to install something it can't be hacked.

Your phone can't be hacked unless you have jailbroken it or let somebody else have access to the phone or your password. You can get malware if you are not careful what sites you visit.