The long wall of text below aside, from what i've been able to read so far
you will want a mingw built version of awesomium before doing anything more.
I found that awesomium has an .so version, which means it definitely could be built somewhere for mingw too.
But it looks like it's not. It's built only for vc++ on windows, which means you cannot use it with MinGW.
At least not easily.
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An .a (archive) is sometimes just a collection of object files, while .dll and .so are shared libraries.
Finally, .lib files are _also_ .a files, as it's the same format
If you have a .dll file you will want the entry points to be able to interact with the shared library
Do you have a .dll with exposed entry points?
gendef my.dll <-- generates my.def
dlltool -l mydll.a -d my.def
Now you can -lmydll with g++ and gcc, as long as gcc finds it
Reading about awesomium, it seems to be in .dll and .so formats
which is windows and linux respectively