Is escritorio not a special office desk or writing desk? Escritorio sounds to me like the German Schreibtisch.
No, it is more general and can even mean desktop or office.
If large was not accepted, you can try reporting it, but it depends on what the rest of your sentence was.
“Huge” would be translated as “enorme”, “inmenso” or “gigante”.
Shouldn't it be "There's a big desk in the office" than "The office has a big desk"? The desk exists in the office. It's not a property of the office, is it?
Well, it is a common expression in both English and Spanish. It may be that the office comes with a big desk rather than a big desk just happening to be there. It is just a different way to look at it. This reminds me of an apartment which comes furnished. When you rent it, there is already furniture in the apartment and you are renting the furniture with it. You cannot move the furniture out to put other furniture in unless you put everything in storage and restore the furniture when you leave and actually you need permission to do that. Really, you don’t usually rent an office with a desk in it, if you have a better desk that you want to use. Places usually cost more when already furnished.