Is the genderless pronoun "they" appropriate and grammatical for a non-binary gender?

I think the reason for your friend's preference is that using either the male or female pronouns implicitly pigeon-holes the person in question as either one or the other. However, all of the examples you give seem to me to be forced, and to shout out loud "Hey, look at how sensitive I'm being! I'm not calling Kris either male or female!"

There are sensible alternatives to all of these examples that do not break any grammatical rules.

"Kris left their umbrella at our house" : Kris left an umbrella at our house (yes, it could technically then be somebody else's umbrella that was left behind, but would you really know?)

"They [Kris] would like more cake." : Kris would like some more cake.

"Kris would like some more cake, can you please pass it to they?" : Could you please pass the cake? Kris would like some more.

While my dictionary (Chambers 1990) does have a secondary definition of "they" as "he or she", it also says that this usage is "with pl. verb", as in "there are lots of people; they are happy". "They is" would not be correct.

I don't know anybody who declares themself to be a non-binary person or genderqueer. In fact, I'm not ashamed to admit I had to look up those terms. But if a person prefers to be not categorized as being one sex or the other, for whatever reason, we should respect their decision. And I believe this is the key issue, if your friend prefers the singular they then use that term whenever they are in your presence.

I'm sure they would be more offended or hurt if you deliberately preferred one gender pronoun, or worse still, used the supposedly politically correct "he or she" as in

He or she would like more cake

Now, how awful would that be?

Themself

^{_{[THIRD PERSON SINGULAR]}} Used instead of ‘himself’ or ‘herself’ to refer to a person of unspecified sex:
I hope no-one else ever finds themself in this position.

They
^_[SINGULAR] Used to refer to a person of unspecified sex: ask a friend if they could help

I nearly forgot.

Is the singular they grammatically correct? Well, yes. Will people object to your using it in speech, I doubt it. We use the singular they in our speech all the time. Would I use it in writing, it depends. If it were a formal letter or paper, probably not. Is the following phrase confusing, insensitive, or ungrammatical?

Kris would like some more cake, can you please pass it to them?

I don't think so, because the sentence begins with the person's first name, which makes it clear you are referring to a friend and to an individual.

^{Sources: Oxford Dictionaries}

I grew up using the singular they, and in my grammar you should not use it for anybody who has been referenced by their name.

So this is wrong:

*Kris phoned, and they said you should return their call.

But this is fine:

Somebody called Kris phoned, and they said you should return their call.

On the other hand, if Kris wants to be called "they", you should try to overcome your natural grammatical instincts and call them "they". Although I would find it somewhat difficult, I would find it easier than calling them zhe or ey.

As someone who is also non-binary and prefers "they" pronouns, the accepted answer is correct, but I'd like to point out that the examples replaced are also perfectly fine (with the exception of the third one, the correct usage there would be "can you please pass it to them").

May I also suggest simply asking your friend (privately) about their pronouns? I'm sure they wouldn't mind, and that would remove any confusion introduced by having a mutual friend tell you about this.