Hell no. I've been in the startup world since 1995, and no one has ever asked me for such a thing.

After 7 years, 33 deals, and hundreds of co-investors, I have never heard of anyone requesting access to an email account. The investor making this request is likely either inexperienced or unreasonable. Probably not someone you want on your cap table.

Absolutely not. This is an outrageous request. If the investors are asking for this now what will be the next request? How much more intrusive will the due diligence process be? This person or persons will likely be overly suspicious and a terrible business partner. My guess is that if the partnership is formed a lot of time will be wasted on things that have nothing to do with furthering business goals.

No. Definitely not normal.

Absolutely not. I would take that request alone as reason not to have any further discussions with this investor. This behavior is indicative of someone who is likely of no benefit and furthermore a potential liability to you and your company.

I've never heard of a request like that before

Who are these people? It is a ridiculous request. You should ask for access to all their personal email accounts and access to their portfolio companies email accounts as a means for you to do due-diligence on them.

As others have stated, this is in my experience, a unique and absurd request that is totally uncalled for and has no precedent.