Can I Still Send E-Mails From My TUM E-Mail Addresses After Leaving TUM?

In general regretfully not. Three month after you left TUM your mailbox will be deleted. After that you cannot connect to your mailbox any more.

Only your e-mail addresses will exist as target for forwarding. This means that you still can receive e-mails that are sent to your / address. You have to configure a forwarding address