I wouldn't share my Gmail password. Besides, I have two-factor enabled on my account, so the password alone wouldn't be that useful.

If you are using Gmail, you can share access to your inbox with another account. This would still let your cofounder access your inbox and read your mail, assuming that's what you are solving for. Your cofounder would also be able to send mail as you, but it will show up as "sent by cofounder@company.com".

I don't even share it with my wife. And the issue is not about trust. It is that there is no need for it. And sharing it with another person makes it a lot less secure.

We share most of the things just because there is 100% trust and it makes things easier. For the specific case of the email we dont access eachothers email, but if there was a need, it would be perfectly fine. I can't say this is the right/safest thing to do with most startups, as it requires extreme trust and you have to know that person better than his/her family knows them.

We have in the past. Eventually we tell each other our passwords after updates. We sometimes have to access something when the other cannot.

Not answering your question but FrontApp.com is a nice solution for those that want to do this.