I don't understand why do we need a 'start_date' for the operators(task instances). Shouldn't the one that we pass to the DAG suffice?
Also, if the current time is 7th Feb 2018 8.30 am UTC, and now I set the start_date of the dag to 7th Feb 2018 0.00 am with my cron expression for schedule interval being 30 9 * * * (daily at 9.30 am, i.e expecting to run in next 1 hour). Will my DAG run today at 9.30 am or tomorrow (8th Feb at 9.30 am )?
Regarding start_date on task instance, personally I have never used this, I always just have a single DAG start_date.
However from what I can see this would allow you to specify certain tasks to start at a different time from the main DAG. It appears this is a legacy feature and from reading the FAQ they recommend using time sensors for that type of thing instead and just having one start_date for all tasks passed through the DAG.
Your second question:
The execution date for a run is always the previous period based on your schedule.
From the docs (Airflow Docs)
Note that if you run a DAG on a schedule_interval of one day, the run stamped 2016-01-01 will be trigger soon after 2016-01-01T23:59. In other words, the job instance is started once the period it covers has ended.
To clarify: