What happens when two separate processes perform a full backup against the same database?
Will these applications interfere with each other if they are both processing the same database? 


Yes they would affect each other, in terms of blocking, when running "concurrently", the one would block the other. But if they are not running concurrently like you schedule copy_only backup after full backup or before full backup, copy_only full backup is not going to affect full backup and you recovery planning. Basically copy_only full backup just backups up the data without changing the differential base and is not going to affect the restore procedure you have planned.

As a fact a lot people use copy_only backups to get replica of database when its required urgently without affecting the total backup and restore sequence.

Few points:


A full backup would block other concurrent full or copy only full backup.
A full backup would not block transaction log backup if you are using SQL server 2008 and above. But log will only truncate when full backup finishes. Paul has series about backup I suggest you read it.
A differential backup would block concurrent differential backup.
A log backup would block concurrent log backup.
If you are using SQL Server 2008 and above you can run transaction log backup when any of full or differential backup is running.


This blogs.msdn article has details


  but what I need to know is whether either backup will not complete, integrity be compromised etc. I need to understand both scenarios namely they run in these orders:


If you are running concurrently the one started first will block other and the other will finish after the fist one allows it to start. But at the end both will finish

If you are not running concurrently, both will just finish fine. Unless there is some other issue.

In both cases the integrity of backup is NOT compromised.