Can a third party conduct an interview using 37 CFR 1.34?

No, a third party or its representative cannot use the provisions of 37 CFR 1.34 to conduct an interview or take other actions not specifically permitted by the rules of practice. MPEP 405 states:

The use of the provisions of 37 CFR 1.34 by a third party or its representative to conduct an interview, or take other action not specifically permitted by the rules of practice in an application for patent, is considered a violation of 37 CFR 11.18 and may result in disciplinary action if done by a practitioner.

This means that attempting to use 37 CFR 1.34 for unauthorized actions could lead to disciplinary consequences for registered practitioners.