Does iOS send contacts to Apple servers? When you go to a contact, you can see if that contact has FaceTime.

How does the system determine this capability?

The usual way that other systems like WhatsApp do it is by uploading/synchronizing your entire address book to the operator's cloud servers, or at least phone numbers / mail addresses (some promise not to save it, but still upload for checking, like TextSecure by Twitter).

Does iOS send your contacts to Apple, especially when you enable FaceTime?

I know it does if you enable iCloud sync, or Siri "for better recognition", as they say, but maybe just using iOS by itself without iCloud already means it will be shared with Apple services.
In some instances, yes.  In the Apple privacy policy breakdown, available here, you can see that Apple indicates your contacts are sent to their servers when using Siri:


  The longer you use Siri and Dictation, the better they understand you and the better they work. To help them recognize your pronunciation and provide better responses, certain information such as your name, contacts, and songs in your music library is sent to Apple servers using encrypted protocols.


However, and this is an important point, all the data is encrypted.  Further in that same policy statement is this:


  That said, Siri and Dictation do not associate this information with your Apple ID, but rather with your device through a random identifier. Apple Watch uses the Siri identifier from your iPhone. You can reset that identifier at any time by turning Siri and Dictation off and back on, effectively restarting your relationship with Siri and Dictation. When you turn Siri and Dictation off, Apple will delete the User Data associated with your Siri identifier, and the learning process will start all over again.


So you have the ability to control what info is sent to Apple, how they use it, as well as being able to reset your unique identifier.  

I would strongly suggest you visit the Apple privacy policy site to get a better understanding of what data is sent to Apple and how they use it.  You can compare this to other mobile companies to see which one takes better care of your data.  I think it's great that you're interested in the privacy of your data.  It's also great that some companies put this info our there for everyone to see while others don't.