Learn Farsi: لطفاً فرم مدیر را امضا کنید.

The verb "to sign" is emza kardan, which is a compound verb made of emza (signature) and kardan (to do). In modern Farsi, compound verbs that use kardan usually drop the be- prefix in commands. So, you say emza konid instead of emza bekonid.

Why does the object marker ra (را) come after the manager, if it is the form being signed?

The word ra marks a specific direct object, but it always attaches to the very end of the entire noun phrase. The whole chunk receiving the action is form-e modir (the manager's form). Because you are signing that specific form, ra waits until the end of that whole phrase.

Where is the word for "the" or "'s" to make "the manager's form"?

Farsi doesn't use an "'s" for possession. Instead, it uses the Ezafe grammar rule. By adding an invisible short "e" sound to the end of the first word (making it form-e modir), it links the item to its owner, literally meaning "form of manager". Farsi also doesn't have a word for "the", so it is simply understood from the context.

What is the little double-dash symbol at the end of lotfan (لطفاً)?

That is a special spelling marker borrowed from Arabic called a tanvin. When you see it over an alef letter (ا) at the end of a word, it makes an "an" sound. This is why the word is pronounced rather than "lotfa".