Both examples are wrong grammatically. You need to ask,
"How much gold does this ring have?"
does is the right word to use here and not do because your verb has to agree with the subject.
Since your subject is the ring, the correct verb to use is does.
Refer this link for more examples and explanation.
Based on OP's updated second question, am making an edit here.
Remember this basic rule
Subjects and verbs must AGREE with one another in number (singular or
plural). Thus, if a subject is singular, its verb must also be
singular; if a subject is plural, its verb must also be plural.
Do matches all plurals as well as the singular subjects I and you.
Does is for all other singular subjects. That’s the system for most
present tense questions.
Similarly, the verb do appears as does if the subject is a singular
noun or, once again, a third-person singular pronoun (he, she, it):
Gus does the housework.
If the subject is a plural noun or the pronoun
I, you, we, or they, use do:
Gus and Merdine do the chores together.