Hello. : ) I hope my answers can be a help a bit. :3
First of all, the question you are asking are covering from beginner - intermediate course, and especially on grammar. So my explanations might be a little bit difficult. But I will try my best to make it simple. So let's start. :)
To 'conjugate' the verb, there is a grammar term that is called 'Attachment / Endings' in Korean. Those two have different names but basically, they are meaning the same thing.
These "Attachment / Ending" is attached to the verb and conjugate verbs into a conversational forms. It is just a theory, so I will explain these attachments and endings one by one.
So, the ones that you've asked are " -다 / -는지" and '-니다 / -요' are the things that you already know. But there is a slight thing that I should point out before I explain things that you've asked.
1. - 다 (O) / - 니다 (X)
'-니다' is not a term to conjugate a verb. There is a '-다' in Korean, and it is a ending form to make a 'basic verb' like 'to-' is in English. So, to give you an example...
In Korean : 가다 = In English : to go
In Korean : 먹다 = In English : to eat
So, '-다' is an ending as It is basically showing a verb as a basic form.
When you conjugate, you need to delete '-다' and add endings and attachments to make it into a sentence.