GCP cloud shell disconnecting frequently

There are several Cloud SQL issues that can cause Lost connection to MySQL server at 'reading initial communication packet'. The most common are:

• Connecting from an IP or GAE app that is not authorized on the ACL.
• You have reached one of the connection limits.
• Your instance has an activation policy of NEVER.
• This can also temporarily happen while an instance is being restarted.

Google offers Container-Optimized OS which is optimized to run Docker containers however this is still in Beta phase. More information about it can be found in this Help Center article.

However, for the Cloud Shell usage limits, as Carlos mentioned in the comment the following documentation explains the usage limits but doesn't provide specific threshold and if your session is terminated or cannot be established because you exceeded a service limit, Cloud Shell will display an error with a link to a form that allows you to appeal the limit violation. Click the feedback link and submit the form with more information about the tasks you were performing before your session was terminated.