First of all - Answer to your 3rd question - No, a cat can't live a normal life it has Rabies. Cats will typically die within 10 days of getting infected with Rabies virus.
2nd question - Yes symptoms come only after rabies infected animal bite.
3rd how to diagnose it, is veterinarian job. Because Rabies symptoms are variable. It has various stages and presents with a range of severity.
Next Steps
For deeper understanding of inquisitive minds-
Early signs: may include behavioral changes, such as anxiety, restlessness, and apprehension. Cats might also show fever and decreased appetite.
Second Stage: Cats may become very aggressive, irritable, and show hypersensitivity to stimuli. They might exhibit erratic movements, excessive vocalization, and a tendency to bite.
Final Stage: This stage involves paralysis starting from the hind limbs and progressing to other parts of the body. Cats may have difficulty swallowing, drooling, and respiratory distress.
Health Tips
Remember- Rabies symptoms vary from individual to individual. All signs in a given subject, at a given time are a rare phenomenon.
Rabies is an EXTREMELY SERIOUS DISEASE. Do not hesitate to take a proper veterinarian consultation in case of animal bites. Hope this helps. Thanks