How to tell if the gas fire is fitted correctly

The advice your home inspector gave you is ridiculous and he should have his license revoked. Either a flue is safe to use with wood, or it is not. Get a good chimney cleaning company that also inspects flues to clean it well first and run a camera down the length looking for any problems that might need attention and assure you it is safe for a wood fire. Don't guess at this or just try a "small" fire before getting a real inspection. Better safe than sorry.

Looks like you are missing one or more parts.

Permanent Attachement

For permanent attachment, most fireplace screen/doors utilize some sort screw (masonry)to attach to the surrounding masonry around the fireplace opening. Usually using supplied screws it would be attached to the opening using masonry drill bit.

Attachment

Other fireplace screen/doors use pressure points on the four corner's to secure (not attach) the door to the opening. Basically the "screws" are turned until they make contact to the opening. The screws usually have a plate or flatter contact point as they do not want to go into the material, merely apply pressure to it. This is repeated 4x for all for corners until it is secure.

To simulate this method of attachment, stand in a door opening and put your arms out and press against the sides of the door. If you apply enough pressure, basically you can hold yourself in place. This method of attachment is not permanent, if one dials back the screws, the door can be removed.

You may be able to pick up some masonry screws and secure it, although I have not seen the back of the door for possible attachment locations or points.