No, you do not need to wait for the Status Icon to be updated within View Superannuation Contributions before completing EOFY. If there is an issue with your lodgement, correcting it is not possible once EOFY has already been completed.
No, you do not need to process EOFY BEFORE processing the first pay run in the FY. This is because e-PayDay Go automatically stores the pay run values against the relevant Financial Year based on the Pay Run Pay Day Date.