Are there any key exchange schemes that only require entropy on one side? Are there any cryptographically secure (in terms of confidentiality) key exchange schemes which are usable if one partner in the connection is entirely deterministic and has absolutely no entropy? I know the basics of the operation of several asymmetric ciphers but am not a cryptographer. I wonder this because it seems like it would be easy to provide initial entropy to embedded devices through an encrypted connection without running into the chicken and egg problem of needing entropy to securely connect to a service to retrieve entropy. If this is possible, is authentication also possible?
No.

A deterministic device initially holding no secret information can no establish any secret information thru public channels.

The argument is elementary: that device can be fully simulated by an adversary, which can thus determine any information that device may contain.