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?
Yes. Many sites will reject mail from you if your mail servers IP address does not resolve to a hostname.
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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.