If I get an email request to do something, what would be some standard polite replies to the effect of "I've just done it" (what you asked for)?
Would
Lieber X,
habe ich gemacht.
Viele Grüße,
Y
be correct? Would it be too informal or rude?
There is no standard polite reply. The level of formality really depends on your company and with whom you are communicating.
In my company the answers range from
Erledigt.
Gruß, Y
(inhouse communication)
to
Sehr geehrter Herr ...,
ihre Anfrage/ ihr Fall wurde abschließend bearbeitet.
Mit freundlichen Grüßen, Y
(communication with extern)
Often the subject line of the e-mail is adapted, too:
Re: Anfrage zu Text - erledigt
Regarding your message "habe ich gemacht." is grammatically correct, but "habe ich bearbeitet" or "ist erledigt" sounds more professional.