No. That is "buzlu çay." In larger cities, some people even say "iced tea." It is not very common in Turkey though.
Could this also mean "the cold farmer drinks tea"? Or would that be "soğuk çiftçi çay içer"?
why not '' the farmers drink cold tea '' ?
If the subject is plural, you will have to use the plural suffix. Your sentence would be "Çiftçiler soğuk çay içer."
Shouldn't "farmers drink cold tea" be ok though? It's marked as incorrect. I think I read somewhere that the unmarked noun in Turkish can be interpreted in a generalised way that is represented by plural in English.
That is never the case in the subject position. Plural subjects are always marker for plurality. It is only true for general direct objects :)