We had a conversation on here a couple of years back when I (at least, I think it was me) referred to a Chinese takeaway as 'having a chinky for dinner'. This offended one of our members (someone I also consider a friend) who has a Chinese wife, so I removed it. I've not used that term on here since, as I don't think it's necessary when I can use other terms and if it means not offending a mate, it's not exactly a hardship.
That said, in the comfort of my own home when discussing meal options on a Saturday night with the wife, I will use that word. I wouldn't call it to someone's face ever, and I'd like to think I have nothing against the Chinese in general as I was once engaged to a girl from there myself (but yes, that falls into the 'I can't be racist as I have a black mate!' argument of crapness, I know), however as a word to use to describe some grub then I'm comfortable with it. I also accept that others are not.
As for kids singing happy birthday in that sing-song faux-Chinese, even I find that a bit close to the edge. Maybe I'd laugh uncomfortably at it if I was there and heard it for the first time, but sat here in front of my PC it doesn't sit well with me.