I want to open a design room, design a mobile phone case and then find a factory to produce it. Can such products be sold in the market? Is it illegal?

As long as the general design does not involve the product patterns registered by others, it will not be sued for infringement. If it is mass production in a factory, I suggest you register a trademark right first, just in case. People won't sue you for screwing around. Everyone in this society has it. If you don't apply for a patent when you grow up, others will sue you for infringement. I hope I can help you.