The word "farm" is directly transfered to greek as "φάρμα" with an "α" to make sound greekish. το αγρόκτημα is the older one and more traditional or greek word. It has almost the same meaning. To that i have to add that George's Orwels "Animal Farm" is traslated as "Η Φάρμα των Ζωών" because it translated from English thus the meaning is closer the original. So if you want to translate "farm" from english to greek "φάρμα" is the safest bet.
Φάρμα is adopted. ;)
I believe most (if not all) native Greeks know the word καλαθοσφαίριση, it is just too formal for everyday use. It is used in appropriate contexts, though, such as in the timetable of a sports event. Furthermore, καλάθι means basket, but μπάσκετ and not καλάθι is used for basketball (what a mess!).
Regarding the first post, το αγρόκτημα, τα αγροκτήματα (the accent moves in the plural so as to not be more that two syllables away from the last syllable).
Do you happen to remember any of those compound words you used to make up? It might be fun!