Learn English – Should I use “Does” or “Is” in this question

Short answer

You should use is.

Long answer

First, let's look at a basic (declarative) clause:

[Presenting yourself confidently]_subj is [a sign of brilliance]_comp

This is called a copular clause. In this kind of clause, we use a form of the verb be to link a subject to its complement. The word copular means 'linking', and the English verb be is the copula, used to link subjects to complements.

In particular, this is an ascriptive copular clause¹, in which the complement tells us something about the subject ("ascribes" a property to the subject). We need a form of be for this to work.

The verb do cannot be used for this purpose:

^# [Presenting yourself confidently]_subj does [a sign of brilliance]_comp

I've marked this sentence with a ^# symbol to show that it doesn't make any sense. The verb do is not the copula, and with do the sentence is nonsensical.

Things get a little more complicated when we consider forming a question. To form the sort of question you're asking about, we need an auxiliary verb to swap places with the subject. The verb be works great for this, as it has its own special grammar: be is almost always an auxiliary verb, even without any following main verb, and so we can form a question by inverting it with the subject in this example:

[Presenting yourself confidently] is a sign of brilliance. 　←　statement
Is [presenting yourself confidently] a sign of brilliance?　←　question

Do can be an auxiliary verb too, but as an auxiliary it requires a following main verb. Even if we wrote the declarative version with do, we wouldn't be able to invert it the way you suggested:

^# [Presenting yourself confidently] does a sign of brilliance.　←　nonsensical statement
*Does [presenting yourself confidently] a sign of brilliance?　←　ungrammatical question

Does here can't be an auxiliary because there is no following verb, and so it can't invert, which means does is both nonsensical and ungrammatical in this position.

To make it grammatical, we would have to add the auxiliary do and invert that. In the following example, the first do (in the form does) is an auxiliary, while the second do is a lexical verb³:

^# [Presenting yourself confidently] does do a sign of brilliance. 　←　nonsensical statement
^# Does [presenting yourself confidently] do a sign of brilliance?　←　nonsensical question

Now the example is grammatical, but it still doesn't make any sense, because do is the wrong verb for a copular clause.

Now let's take a look at your full sentence, which we can regard as a reduced form² of the following:

*Does [presenting yourself confidently and believing in your own ideas] a sign of brilliance
　or
does presenting yourself confidently and believing in your own ideas a sign of ignorance?

This is both nonsensical and ungrammatical. However, with is it's grammatical and makes sense:

Is [presenting yourself confidently and believing in your own ideas] a sign of brilliance
　or
is presenting yourself confidently and believing in your own ideas a sign of ignorance?

It's grammatical because is is an auxiliary and can invert with the subject, and it makes sense because is is the right verb to use in a copular clause.

Notes:

　¹Also called a predicational copular clause.
　²Basically, the repeated material has been deleted.
　³Verbs can be either auxiliary or lexical; auxiliaries can be modal or non-modal.

Symbols used in this answer:

　^# - nonsensical (wrong in terms of meaning, not grammar)
　* - ungrammatical