If the code is working then I would vote against any major automated rewrite.
Lots of work could be involved fixing it up.
Small iterative improvements over time is a better technique as you will be able to test each change in isolation (and add unit tests). Anyway your major complaint about not being able to find the code is not a real problem and is already solved. There are already tools that will index your code base so your editor will jump to the correct function definition without you having to search for it. Take a look at ctags or the equivalent for your editor.
Messy
Subjective
Makes it hard to find the implementation of a method (especially searching through a tree of classes for a virtual function, only to find one class had its version declared in the header...)
There are already tools available for finding the function. ctags will make a file that allows you to jump directly to the function from any decent editor (vim/emacs). I am sure your editor if nto one of these has the equivalent tool.
Probably increases the compiled code size
Unlikely. The compiler will choose to inline or not based on internal metrics not weather it is marked inline in the source.
Probably causes issues for our linker, which is notoriously flaky for large codebases. To be fair, it has got much better in the past few years, but it's not perfect.
Unlikely. If your linker is flakey then it is flakey it is not going to make much difference where the functions are defined as this has no bearing on if they are inlined anyway.
