Is it also possible to have "Czy ty konie kochasz"??
„Czy ty konie kochasz” is also gramatically correct, but it stresses in a non-neutral way. Perhaps you shouldn't use it until you are more advanced.
As Vengir said – grammatical, but too unnatural to accept it. Saying this instead of “czy ty kochasz konie” is like saying “Horses – you love them?” instead of “Do you love horses?”.
"Czy ty kochasz konie" would be OK.
Is it also possible to have "Czy kochasz konie?"
(So, drop the pronoun, since that's usually okay, but still have czy to make it a question?)
That's right. You can also drop „czy”, as long as you use the proper intonation.