Will this RedGate backup plan interfere with a new differential backup plan I want to create?
If I setup a differential backup plan and someone comes along and
  manually creates a backup without specifying COPY_ONLY then it's going
  to throw the plan off, right?


Yes, your next differential backup will have a differential base LSN of the full backup the other person performed so your restore chain would require that backup.


  Secondly, we currently have a RedGate backup solution in place that
  uses the code below. I'm not familiar with how the master..sqlbackup
  proc it uses works so I am wondering if it's possible for me to leave
  this in place if I setup a differential plan alongside it or if I
  would need to disable this.


If you add a differential backup in parallel to this, your differentials will still require the RedGate backups in order to restore the DIFF because the RG backups will reset the differential base LSN.


  Why would I do this? Well, we're migrating to the cloud and I'd like
  to setup differential backups for the purpose of minimizing downtime
  as we move the database. But I'd just as soon not mess with the
  existing backup plan if I don't have to. And if I have to then I have
  to, but would rather not just in case something goes awry with my
  backup plan in the process.
  
  EDIT: And I can already imagine someone might ask why I don't just use
  RedGate to do the diff backup plan. First, I have no experience with
  it and second we won't have RedGate on the new server.


If you don't have RedGate on the new server, how are you planning to restore FULL the backups taken by RedGate in order to restore your differentials? My understanding is that restoring a backup taken by RedGate requires RedGate to be installed. I could be wrong (limited experience with RG) but this is how most third-party backup tools for SQL Server work, usually because of custom encryption or compression that is applied to the backups.

Without additional info available, I would suggest one of the following:


Replace RG on the existing server now with whatever you're planning to use in Azure as an end-state (unless you're using Managed Backups in Azure).
Look at Log Shipping to keep your Azure server in sync with on-premise until you're ready to cut over. You would have to disable any RedGate log backups that are in place.