Paint.net does not by default display the mipmaps, I don't think you need to ever see the mipmaps anyway. I don't think you can even get Paint.net to display the mipmaps.
Paint.net apparently creates 9 mipmaps when that setting is enabled in the save-as dialog for DDS files. That's fine.
There is an excellent tool for viewing and converting various graphics formats, it is http://www.btinternet.com/~mnwright/. On that home page look for the button for DXTBmp on the bottom of the page, you also need the "GFX DLLs" package (on another button). In Win 7 and Vista DO NOT install to the "Program Files" or "Program Files (x86)" folder, I install this kind of stuff to a "Utils" folder to avoid UAC problems. The DLLs package will install to the System32 and on an x64 Windows to the SysWOW64 folder, that's fine. The DXTBmp utility can load a DDS and "Preview" the mipmaps including identifying how many mipmaps are there. DXTBmp will also load a file in some other format, like .bmp or .tga, and convert it to DDS. It's a very useful graphics tool.
An experienced graphics person
might want to edit mipmaps directly to improve their quality or remove artifacts (I have occasionally needed to do this but it is rare). That's the only reason you might want to see and edit mipmaps directly. Ordinarily you just let the graphics program create the mipmaps and assume they will be fine.
Mipmaps are used as alternate textures depending on the distance that object is viewed from. When the object is viewed from farther away the full detail of the texture does not need to be shown, progressively lower detail mipmaps will be used for the texture as the distance increases. This is a graphics optimization which increases graphics performance. LOD, Level of Detail, settings in a graphics engine control which mipmap will be used at what distance. It is not strictly necessary to have any mipmaps at all, most graphics engines will resize a texture but this will lower graphics performance and may introduce texture artifacts therefore mipmaps are a better solution.
Will I end up with a single .dds file as before?
Yes, the mipmaps are just stuck onto the end of the file, they are part of the graphics file.
Something you haven't mentioned yet: Does your jacket have a normal map file?
If your main graphic file is named "jacket.dds" you might have a "jacket_n.dds", that's the normal map file.
That's a separate graphics file which is usually present but not strictly required. The normal map has two channels, the first maps the lighting of the object, the alpha channel maps the 3D texture. If your jacket has a normal map file you probably do not need to change anything in it as a result of adding your smiley face emblem. You could give the smiley face a 3D effect by modifying the normal map file but it's not necessary.
If your friend who created the jacket did not also provide a normal map you might ask if that could be done. Creating the normal map and bump-mapping is not simple and is a job for someone who has some graphics experience. Again, I don't think the normal map file is strictly necessary. Try what you have in the game and see if you get the results you wanted.
Just a note: Unfortunately it is almost impossible to do alpha-channels in Paint.net, it is possible but extremely complicated. I gave up trying to make Paint.net work with alpha channels.
Another note: The GIMP is an alternative free graphics editor. I do not like GIMP, it's not intuitive at all and is unlike any other graphics editor but it's lightyears more capable than Paint.net. One major failure is that the current version of the GIMP is not really supported for Windows and worse it may not properly save DDS files. A workaround for the DDS issue is to save a file in some other format, like .tif or .tga, then convert it to DDS with an external converter like the nVidia converter included in the free nVidia DDS utilities package.