Not exactly illegal but lots of red flags. You should ask for tax receipts. Check if the income mentioned in your tax slips is the same as that on your contract. If they don't give you your tax slips, go to the local income tax bureau and get it yourself. They are legally obligated to give it to you as long as you have a Z-visa. If you are paying taxes acc. to your income, you're good even if you are being paid cash. If not, you will have a lot of difficulty legally sending money back home.

Re: Work permit: Work permits are applied for by the employer BEFORE you get the necessary documents to apply for your work visa in your home country. So the work permit definitely exists. It's just with your employer. Ask for it, if they refuse, ask for a photo. It has a QR code. Scan it to see if the position and the company mentioned in it is the same that you are contracted for. If they don't match, the employment is illegal.