Is it okay to have a white logo on the navigation because of a darker colour shade and the original multicolored logo in the footer? Is this an accepted practice?
Yes it is ok. Having different versions of your logo is more than accepted practice, it's recommended. Large companies will have a Graphic charter, which are brand guidelines that details precisely how to use their logo in different situations, for example on a light or dark background, very small or very large, which precise colors to use and much more.
They are usually available online, for example you can find the Twitter guidelines here.