The embassy of the United States in Denmark offers complete information about marrying in Denmark:

Be lawfully present in Denmark
Documentation showing present marital status, i.e. copy of your last U.S. tax return.
If you live outside the U.S., present a valid residence permit from your country of residence
Valid passport showing proof of entry into Denmark or into the Schengen area
If you have been previously married, you must present your original divorce decree.

In order to get married in Copenhagen, you need to personally visit the Citizens Service Center (Borgerservice) at: Jarmers Plads 7, 1551 Copenhagen V. Borgerservice cannot be contacted by telephone. www.kk.dk If you are planning to get married in Copenhagen, but you are not yet in Denmark, you may contact the Copenhagen City Hall as follows: telephone: +45 33 66 33 66. E-mail: borgerservice@kk.dk

Non-residents pay a fee of 500 Danish Kroner (DKK) to get married in Denmark.

As to your other questions, where exactly are you settling? France? You could start with the Expat guide to Paris to give you in-depth information about moving to the city and the visa process. You will probably need to apply for the Single Permit Directive which allows immigrants who are legal residents of an EU country and those who are seeking residence permits to apply for a residence and work permit for that country through a single application process.