"Directorate General of Passports" is the most-used of the three terms you suggested.
It is used in non-English speaking countries; it gets most google hits. (1400)

the other two get way less hits.

none of the three terms sounds good in English.

So it's a usage question but not really a native-English usage question (IMO)
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In the US, it's called "Passport Services Directorate"
http://travel.state.gov/passport/about/who/who_894.html

other non-exclusively English speaking countries call it this:

http://www.dgip.gov.pk/
Directorate General Immigration and Passports (Pakistan)

Directorate General of Passport & Residency:
http://www.destinationoman.com/getting.html
Oman

http://www.saudi.gov.sa/wps/portal/!ut/p/kcxml/ndDJDoIwEAbgZ...

http://www.gdnpr.gov.bh/
The General Directorate of Immigration and Passports in the Kingdom of Bahrain

http://www.migration.gov.rw/services.php
Directorate General of Immigration and Emigration

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Note added at 1 day14 hrs (2008-04-20 19:38:13 GMT)
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Answer to your first question: Directorate General of Passports is not wrong grammatically, but it sounds very informal, not professional. It sounds like an office where passports are physically located but it does not express the "function" and importance of the office although this should be conveyed through the title. The "function" is issuing etc. of passports, in other words, passport services.
I would say it's okay to call it "Directorate General of Passport Services" because it includes the function and that makes it sound professional and good.
To your other question: General Directorate of Passport Services would be grammatically correct also, but it again does not imply the official statewide authority/importance of this office that is always expressed with the capitalized "General" following the "Directorate", as also in "Consulate General". General Directorate sounds like there is another more "specific" directorate.That's all.
I would also capitalize the other nouns because it's an official title/name.
Hope this helps.

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Note added at 1 day15 hrs (2008-04-20 19:57:35 GMT)
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Other titles also expressing "functions" (although the functions might be different) and sounding correct are:
Directorate General of Immigration and Emigration
Directorate General of Border Security
Directorate General of Passport Control
see also Gary D's comments to my answer.