How to plan for change to log backup schedule?
If I change to every 15 minutes, do I have to set a new start time? 


Nope! The start and end time is when the schedule is active. So, assuming you still want to take t-log backups every 15 minutes all day, then you just need to adjust that value and the when the next hour/half hour occurs it will take the backup.


  So for example, if I set a new job at 10:00am, would I also need to change the start time to 10:00am?


No - because 10AM is still between 12:00AM and 11:59PM, so the schedule will still be active.