1. As an standard response to 'Thank you' or 'Sorry', people use the fixed phrase 'No problem'.
"Thanks for helping me!"
"Sorry I'm late!"
2. We can use 'No problems' in a literal sense. For example:
"What did the garage say about your car? Is it all right?
"Yes, it's OK."
"No problems? So, the brakes, and gear box and everything are OK?"
"They're fine. No problems at all!"
In this case, we are talking about actual or potential problems, so we use the plural.