Where are flaws caught? Is there a QC department in DESIGN of a product or only in the production phase?

If done properly it permeates every stage.

Designing:
Ensure as far as reasonably practicable, that the item will meet its buyers' needs and specifications, meets any relevant national or international standards and regulations - and can be  made efficiently.

Those demands are heavily interwoven, with many customers demanding the goods they pay for will meet all those.

Prototype:
Test to ensure it works as expected and meets the regulations and standards - live electrical or moving parts are properly guarded, for example. Samples might have to be sent - at the maufacturer's considerable cost - to independent test laboratories to verify meeting safety and environmental regulations. (The European Union jargon for such a lab, is "Notified Body".)

Production:
Continue tests to ensure continuing function and compliance.

For example, at a simple level parts are measured to ensure they actually fit together, as a complex assembly often has its separate parts made by many different companies, some in different countries.
At a higher level, welds on critical pressure-vessels are examined by ultrasound or X-ray to ensure they contain no flaws that risk catastrophic failure.  Not only that, but the welders themselves have to have proven professional skills.

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In full QA (Quality Assurance) schemes, it is all recorded, creating heavy-going audit trails from original concept via the raw materials and bought-in parts like screws, to the finished item delivered to the buyer. The paperwork itself may be subject to management-control systems like ISO9001. (That is frequently but wrongly thought of as a product or service quality guarantee. It is not - ISO9001 merely helps maintain consistent quality even if that is consistently low.)

These days too the product may become wrapped up in the ISO1400x-series of internationally-agreed environmental credentials. That controls the manufacturer's behaviour, not the product directly; but many major customers, especially Government agencies and their direct suppliers, now insist on suppliers having that approval too. Among other things it verifies that the factory's waste-products are disposed off in a proper, and regionally lawful, manner.

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The aim is to create as few flaws as possible! No-one would be daft enough to say flaws can never occur, but everyone does their best to minimise that risk, and certaintly to minimise the risk of any flaws creating unwanted hazards in using the product. (Distinct from the known or sensibly expected hazards that is, and you can't protect the users from themselves, of course.)

To acheive that aim, quality-control begins in the designer's office and ends in the finished product, and its accompanying instructions and guarantee.

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You may have seen a curious symbol, "CE", on goods either imported by American companies from the EU, or made in the USA or elsewhere for export to countries including the EU that invented it.

That CE covers a morass of formal QA along the lines I have described. The product may come with a list of the relevant EU and international QA approvals it needs to gain CE-marked release for legal sale "by way of trade" - as the lawyers have it.

If all its approvals are to ISO standards, then provided the labels are genuine and not counterfeit, I would think the product would be legal for sale in the USA, a full member of the International Standards Organisation.