I have never heard anyone say, '...talk til something pops.' If you actually heard this expression in a job interview, what else can you say about it? Did it seem like a good thing? People often use expressions like, 'talk til you're blue in the face,' (talking to someone for a long time without getting the result you want), or 'talk till the cows come home,' (talk a long time.') Talk till something pops, could mean many things.
Here are two 'pops' expressions:
pops into my head=occurs to me as a thought;
I was about to pop=refers to eating so much one feels full to bursting.
Perhaps 'talk til something pops' means, 'talk until something occurs in the conversation that lets me know you are right/wrong for the job.'
But these are just guesses.
To cry oneself to sleep means just that. To be so miserably unhappy that one cries and cries to the point of exhaustion and then falls asleep. The sadness can be from any cause: grief, disappointment, heartbreak, frustration, etc.