How to turn off notification sounds in Microsoft Teams

You are using the click-to-run version of Office 2016. Thus, the Office updates are not controlled by Windows Update.

When automatic updates are enabled for Office clcik-to-run version, in most cases updates are applied automatically in the background without any user input. However, updates can't be applied if an Office program is open. If an Office program is open, other attempts are made to apply the updates at a later time. If, after several days, updates haven't been applied, only then will users see a notification that an update to Office is available.

To disable the update notification, we can try the following registry key settings:

HKEY_LOCAL_MACHINE\software\microsoft\office\16.0\common\officeupdate
Value Name: hideupdatenotifications
Value Type: REG_DWORD
Value Data: 1 

You can also create the hideupdatenotifications value with group policy setting to disable it:

HKEY_LOCAL_MACHINE\software\policies\microsoft\office\16.0\common\officeupdate

Is there is a way to disable the toast notification but keep the flashing toolbar icon intact?

It appears what you want will not be possible.

Based on, Notifications in MS Teams, the following is true.

Here are the results from some of the testing I did in understanding the MS Teams notifications model. I’ve run this by someone on the MS Teams product team and they tell me I'm "absolutely correct".

The user concluded the following:

You cannot turn off notifications for:

• Personal mentions

• Followed channels

You can turn off notifications for:

• Team mentions

• Channel mentions

• Replies to conversations I have started

• Replies to conversations I have replied to

So it appears that toast notifications for Microsoft Teams cannot be disabled, still keep the taskbar icon flashing. It's possible the toast notifications still appearing, in the Activity Center even after they are turned off for all applications, is actually a bug with Microsoft Teams though.