What's "big deal" mean and when's it used?

Is it grammatically right to say just "until you"? Or is anything skipped here?

Yes, there is nothing omitted here. The speaker has used "until you" at the end here to specify that her love was the only one that really mattered to her, but you can use other similar expressions to give the same idea. Ex: I was fine 'cause I knew they didn't really matter until I met you. Ex: Once I met you, I was fine because I knew they didn't really matter.

"Don't worry" here sounds like "Zan worry". Which accent is it? Is it common to pronounce it like this, or is it exceptional case?

Some particular accents from the UK will cause the long "o" sound like an "a" when pronouncing certain words. This may be why you heard "don't" as "dan." You don't need to worry about imitating this accent, as it is not common outside of the UK.

Why "is" were used for two times in here?

It appears the speaker stuttered on the word 'is'. Using 'is' once is the correct form of dialogue.

What does "convert" mean in this context?

To "convert" means to change or cause transformation. So in this context, "convert" means to cause people to change their religious views by accepting the gospel and becoming a Christian.

Would the nuance of the sentence change if I use "I" instead of "we"?

The nuance doesn't change substantially here if you substitute "we" with "I." Some people like to use "we" instead of "I" when asking questions, as it gives more of a feeling of collaboration. However, since the speaker is asking for the woman's opinion here anyway, you don't necessarily need to use "we" here to provide the same effect. Ex: A: I'm starving. B: Should we get you some food? Ex: We should buy you some new shoes. Your current ones look worn out.