May I use my former-yet-active email address of an institute as a contact channel in my current CV?

I can see no legal or ethical problem with using an email address from a former institution, as long as you are clear about your current affiliations in your communications.

That said, there may be better options:

• Professional societies often offer an option of an email address. As a roboticist, for example, you can almost certainly affiliate with the IEEE, whose excellent email service I use myself.
• One of your former institutions may offer an alumni email address (or at least forwarding service). I believe this is more common in the US than Europe, but it's fairly arbitrary which institutions do and do not offer this.

Both of these can typically be set up very quickly and also have the advantage of being permanent, even when you change institutions.

I don't think the premise holds. I have been on the hiring committee for many post docs, and I have never heard anyone mentioning the email address of a candidate. I could just as well cook up an imaginary argument saying that a hiring committee will find it unprofessional that you use your work email to apply for a job (that also does not hold). I would probably not go for wildly inappropriate handles, but normal ones are not a problem.

If you don't like using your gmail in professional settings, set up an alternative. I use [email protected] (which just happened to be available), and it is easy to set it up to forward to gmail. So you don't even need a mail server of your own.

You're overthinking this.

[email protected] is just fine as a professional email address.

When people talk about unprofessional emails it's the bit before the domain that causes issues, not the domain itself (unless you've deliberately gone out of your way to pick an unprofessional one).

If you're unsure, you can always go with firstname.lastname @ CommonlyUsedDomain