Do outgoing SMTP servers need to have assigned domains?
If a domain name is not assigned to an outgoing SMTP server will it increase the spam score?
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They don't need to have them per any standard in existence - however, as any mail admin with any experience would be able to tell you, a vast amount of anti-spam systems do use the lack of a reverse record as a reason for negatively scoring an SMTP relay.
Which means, the answer to the question in your title is - generally, yes.