No, items purchased in the Premium Shop are not included in storylines or quests and are not required to complete missions. Similarly, items featured in the Premium Shop are not part of the core inventory collection and will not be necessary to complete IP-related clothing collections.  
Please note that there is a known issue that currently displays a few Premium Shop items within the Collection Menu. This is a bug that will be fixed in a future update. You can track this issue on our Trello board.