The form -ㄹ 테다 expresses expectation or intention. In practice, its most common usages are in the context of 텐데(요), 테면, and 테니(까). However, as you point out, it can be used more directly in the infinitive form of 테다 (although the other forms are more common in practice). Usually, when it is used in this way, it is the first person imperative.
안 줄 테야 (I will ...
There is no need of using 을 before 하고 in this sentence.
하고 is a postpositional particle, which is attached to a noun or noun phrase. It indicates a behavior is done with someone as a partner.
을 is also a postpositional particle which indicates the noun is used as the objective case of the sentence.
You can only use one postpositional particle after a noun. ...
In my experience, 아/어도 is a stronger or more emphatic form than -든지.
I would translate the sentences you gave into English, with the following nuances of difference:
저는 무슨 일이 있든지 약속을 지킵니다: Whatever happens, I keep my promise.
저는 무슨 일이 있어도 약속을 지킵니다. No matter what happens, I keep my promise.
The second sentence, at least to me, carries slightly more ...
They are about the same in your examples. I think 했다면서요 sounds slightly more informal, but the difference is negligible.
One minor thing is that -다면서요 has another usage while -다지요 (short for -다고 하지요) is used only in this one sense (but it can be used in both declarative and interrogative sense).
영철이는 과에서 여행을 간다면서 일찍 나갔어요. (간다면서 = 간다고 (말)하면서)
영철이는 아침 일찍 ...