You CAN get married while he's there on a visa-waiver BUT the onus is on you to prove to INS that it was SPONTANEOUS rather than planned, and believe me, they'll inquire.......it's a grey area, but possible.
Hypothetical Situation:
Marrying a US Citizen who lives in the USA in the USA in the "spur of the moment" (Visa Waiver/Adjusting Status after Marriage)
You wake up one day during your Visa Waiver vacation to the USA and find that you should marry the man/woman next to you. You have never planned to do so, you have evidence if you would be asked at the port of entry that you don't intend to enter the US to marry. (Appropriate evidence is: Flight ticket back, Evidence that you got a job in Germany, Evidence that you got a house/apartment/property in Germany, Evidence that you have a bank account in Germany, ALL kinds of Evidence that underline the spontaneous idea. If you do not have sufficient evidence it could happen that the future applicant is barred from travelling and residing in the US for 3-10 years and get deported immediately as well as of course not being issued a Visa!
You marry your US fiance(e).
You file form I-130 and I-485 at your local USCIS Office. Always make sure you read the instruction pages thoroughly! In the instruction pages you find which Service Center and local USCIS Office works on your case. I.e. for Atlanta, GA the USCIS Office is in Atlanta.
For work permission file form I-765. Read the instruction pages on where to file.
As initially said, the quickest possible solution for a couple or a family to be united in the USA is the DCF (Direct Consular Filing). There is no other option comparably as smooth and easy. Whether you decide to go for a K1- or K3-Visa might rather be a personal question with personal consideration of plans under given circumstances. None of them is usually that much quicker than the other. Most of the times it depends on the Service Centers and their processing times. The one thing I highly appreciate is that with a K3-Visa you will be able to travel back and forth from/to the US without having to apply for yet another document, the Advance Parole. Last not least there cannot be an advice on the spontaneous marriage, for how could it be spontaneous if considered before...?!