Can tooth cavities be reduced or eliminated using certain toothpaste or remedies?

Possibly. There is no magic paste out there that will completely eliminate to risk of getting cavities. However, any fluoridated toothpaste can help protect your teeth and reduce the risk of getting cavities. If you are a high cavity risk patient, there are special prescription toothpastes you can use.
Incipient. Sometimes incipient proximal small etched areas can be remineralized, if your dentist has them on "watch" it would best for you to have superb homecare, flossing, brushing with ada recommendations, plus use adjunctive like mi paste plus, or carifree system, xylitol, etc. All will help.
No. Once a cavity is diagnosed by the dentist, need to remove with filling, or other alternative treatment based on the size.
No. . No, any cavity that you can see or feel will require a filling. Cavities will never go away or heal themselves, they will only get worse. Fluoride can prevent cavities and stop them from developing in the first place but only before they are evident to the naked eye or you can feel them.
No. Once a cavity develops, the only way to stop its progression is by removing the decay and restoring the tooth.