How to choose which backup files to restore when identical timestamps exist for diff and log backups
After restoring the full backup, do you restore the second differential backup and the last log backup?


This is going to depend, but assuming:


The source server is offline and the data files cannot be accessed
The most up-to-date copy of the data is what is asked to be restored
All backup files are successfully accessible


Then you'd want to restore the 1:00 AM full backup, then the 11:00 AM differential, and there will most likely be some additional information in the 11:00 AM log backup depending on the length of time the differential and log backups took.


  Or do you restore the first backup and all the log backups (even though the 10AM log backup really has the same issue as the 11AM backup - coinciding with the differential backup in each case).


I'm not sure what you mean about the 10:00 AM log backup having "the same issue" as the 11:00 AM log backup. There isn't an issue at all.

The whole point of a differential backup is to apply the changed extents and then keep just enough transactional information to make everything consistent. This greatly differs from a log backup which is the actual log records. They serve two different functions and you don't need differential backups in a restore to the latest point in time if you're in the full recovery model but you do need log backups to do so.


  … do we really have to resort to header restores to determine which files are able to be restored in sequence?


You don't have to "resort to header restores" if:


You have an up-to-date copy of MSDB and can query the metadata out of it
Some other 3rd party application does the metadata shuffle for you
A human sits in front of a screen using SSMS to load up all the files and let the restore wizard figure it out by utilizing the backup header metadata
You keep some other metadata repository that is synchronized with your DR area


If, however, none of those (not complete, just examples) are the case then... yes, you'll need some way of figuring out which files go in which order. Since the easiest way to figure it out is to ask the backup what metadata it holds, that's what you'll find most people doing.