Yes, all emeralds have certificates. Emerald without certificate may be real or fake.

If there is no certification, businesses may not do it, and some very cheap goods may not be certified in order to save costs. You can go to the jewelry testing center to identify whether it is A goods. If it is A goods, then there is no doubt.

You can't see the problem from the picture, but it should be true. Generally, fakes are not sold in that place, but the price will be very expensive.