My study institution claimed that they sent two important mails. I am quite sure I did not receive these emails.
I want to rule out that I received them and just deleted them. Is there any way that I can show that I did not receive these emails?
You'll need to contact your mail administrator and ask them to trace those e-mails. Based on senders and destinations and approximate time gap when they were sent, they can check server logs and see if indeed they were delivered to your mailbox, if they were sent to a different address, etc.
That will depend on your institution policy whether they are permitted to perform such an audit and provide you with logs, etc., though.
Just by yourself it's complicated to prove you did not get those e-mails, server logs will probably provide a more robust answer to this.
No. E-Mail doesn't provide a reliable mechanism for this. You can write mails that "require" a feedback whenever they've been read. You could tell them to show you that they got that information for those E-Mails. But sending and requireing this is an optional feature.