Our copyrighted logo will be included in the repo
This is no bar. All the code you write is copyrighted, and that's not a bar to publishing it - the question is what kind of licence you wish to give end-users. It's perfectly fine to publish the code under a copyleft licence, but allow (eg) only verbatim reproduction of your logo image, and that only when used with unmodified code. Many mainstream open source projects do something like that; firefox (and the consequent iceweasel fork) come to mind. It's probably not a great idea to include a logo if you grant no copying or redistribution rights, though; just leave it out in the first place.
It will depend on an Api that is paywalled - so this application is highly coupled with a closed Api.
I can't quite parse this, because an API would have to be part of the client-side code. I think you mean "this application talks to paywalled network resources", and if you do, it's fine to publish an open-source client whose principal purpose is to talk to your non-free network service. It's very common in the Android world; literally fifteen seconds scrolling through my f-droid turns up mundraub, the first of many free (GPLv3) clients which talk to non-free network services (in this case, https://mundraub.org/).
Why is this desirable? Firstly, because legitimate users of the non-free service might want a free client to talk to it. I can't use the apps of any of my banks, much as I'd like to, because those apps are non-free, and I won't run non-free software. Secondly, because people building comparable apps to talk to other network services can then modify your code instead of starting from scratch, and as long as your code was strong-copylefted everyone benefits from all this work.
If instead you mean "this client would have to use proprietary API code", then it's not a good candidate for freedom.
