Hi VicMill,
Please find the below link for performance issue, please see the recommendations for the project site sync in the first document.
1) https://support.office.com/en-us/article/Tune-Project-Online-performance-12ba0ebd-c616-42e5-b9b6-cad570e8409c
2) https://blogs.technet.microsoft.com/projectsupport/2015/02/03/project-online-some-changes-coming-to-task-sync-for-enterprise-projects/
And for opening projects in read/write mode, in the Browser >> PWA >> click on server settings >> Queue And Database Administration >> Force check-in the Enterprise Objects >> and select the projects to be checked-in and click
on Check-In.
If PWA check-in doesn't help, be cautious doing the below activity, prior doing this in the Project Pro, check if there is any offline data has to be sync to the server, if you cleared this your offline data will be lost. I will do this monthly once or if
I reached a deadend point.
1) Go to Project Pro >> Connecting to your server
2) Click on Options >> Save >> Click "Clean up Cache..." Button
3) Select the Appropriate "Project Filter" and click on "Remove From Cache"
4) And now try to open the Project plan in the Browser or Project Pro...
Hope this helps you...
-SNKBalaji