I will try to answer your questions at a high level, and I (or others) can add detail as you need.
1. Jenkins and NiFi can both use CRON scheduling to start processing.
2. For any failure of a processor in a NiFi flow, you can route to a processor such as PutEmail to notify operation team members.
3. NiFi offers Data Provenance through which you can track how data flows through NiFi. Data Provenance provides a way to search the Data Provenance repositories for flowfile names, component IDs, and other attributes to determine what happened during the data flow. Jenkins has a more specific status display for jobs, but NiFi's Data Provenance UI may also serve your purpose, though it is more oriented towards data flow.
4. Data Provenance, described above, allows you to track the status of data flow in NiFi.
Do these answers help you, and are they detailed enough?