Hello,
Keep in mind that help you depends on elements you provide so, more you give us better we can help you.
I ask for a step by step that can able us to see how you test your plugin, your answer is just we're agree the classic way, but there is other ways.
Other strange point, in your log some of your test show that some notification from Shipstation server were without username or password parameters ! Don't know how you can have this kind of results...
But from what we can see, I notice something you set your Shipstation plugin to export only for "shipped" status while on your order listing your order are only in "confirmed", and so that's normal that nothing is exported.
Regards