Are you using on-prem Exchange server or Exchange online?
1. The problem is I can't see the members in Outlook client.
Yes, that's by design. It is not expandable because its membership is calculated when a message is sent to it.
However we could use command to view the users in DDG.
Get-Recipient -RecipientPreviewFilter (get-dynamicdistributiongroup <name of group>).RecipientFilter -OrganizationalUnit $group.RecipientContainer
2. Is there any way I can use PowerShell to create a script to automatically assign a Distribution Group to a new user?
If so, you will need a powershell script that creates your accounts and make adding that distribution group part of the script, and please note that we help users modify their scripts when they have an issue with it, but we don't support for writing scripts directly.
3. upgrade the Distribution Groups to Office 365 groups and use dynamic membership rules?
You could take a look at the operation introduced in the official document: Create or update a dynamic group in Azure Active Directory
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