Should I use the Reply-To header when sending emails as a service to others?

Question

Suppose we have an application that acts as a middleman, allowing Company A to send reports to their customers.

Company A --> Company B (me)--> Compan

Answer 1

I tested dkarp's solution with gmail and it was filtered to spam. Use the Reply-To header instead (or in addition, although gmail apparently doesn't need it). Here's how linkedin does it:

Sender: messages-noreply@bounce.linkedin.com
From: John Doe via LinkedIn 
Reply-To: John Doe 
To: My Name 

Once I switched to this format, gmail is no longer filtering my messages as spam.