Please tell me, could one have a small tooth decay but without bad breath or saliva?

Absolutely: Small cavities do not cause bad breath. They do turn into larger cavities requiring root canal therapy or extractions if you just wait and do not get them fixed. When was your last dental check-up and cleaning?

Definitely, yes: However, the proper diagnosis is the key to a successful treatment. See your dentist for Halimeter test, diagnosis and bad breath treatment that depends on its cause. Good luck.

Yes: Yes. Have tooth fixed. Early sign of tooth decay is any dental pain.