Safe delete Transaction Log in SQL Mirroring scenario
Transaction log is the most important element in SQL Server. It is like a sequential journal that logs all the changes made to the database and contains enough information to undo / redo the changes in the event of crash recovery to keep the database in a consistent state.

Highly recommend to read : Understanding Logging and Recovery in SQL Server

All your question are just because you dont have proper understanding of how transaction log works.

To keep transaction log in shape, you should take regular log backups (more frequently) on the principal server. Check - How do we handle transaction log maintenance for a mirrored database?



To answer your questions :


  Would it be safe taking full backup of Principal Server and then delete the transaction log completely?


No this is not possible and not advisable to delete T-Log.


  Does it have to do something with Full Recovery/ Simple Recovery Mode?


Mirroring uses T-log to replay everything from principal server to mirror server. So Full recovery is only possible. check : recovery models.


  What would happen when the transaction log on the Principal Server is deleted and the after sometime Mirror Server will take the place of the Principal?


Not feasible. Please read the links that I have suggested.