Thinking about it, in my circle of friends, we will use the terms 'yellow' and 'brown' if it relates to us an individual. I suppose it's a bit like how black people use the n word but no one else does - it's about reclaiming a word. It's OK for you to use it, but no one else. So if you're a Caucasian person, steer clear of colours. Especially 'coloured people' and 'persons of colour' - those are just awful terms.
Personally, I hate the term Asian because people in this country use it interchangeably for Indian (what about the Chinese people, eh?) and I don't have a lot of love for the term BAME because people in this country use it interchangeably for black (what about the other ethnic minorities?) so the terms as inoffensive as they are have gained offensiveness for me due to how they've been repeatedly misused.
I also hate the term 'dual heritage' - that implies on an upbringing to two cultures based on the way you look. Your upbringing may be based on only one of those cultures or a different culture altogether. Don't make assumptions.
Mixed race is fine - it's factual. It doesn't imply and doesn't judge.
'Half-caste' or 'mixed caste' is just deliberately offensive. No.
In a slightly more pop culture reference, 'mudblood' is OK if used by someone who is mixed race to describe themselves, but not if used by other people. I refer back to my comments on 'brown', 'yellow' and the n word. You can use any term to describe yourself, you just can't impose it on other people.
Ultimately though, as someone who isn't white, what it really comes down to is meaning. Some terms are racist, some aren't - but I can hear in your voice and see in your body language if you're trying to upset me, if you think I'm worth less than you, or if you're using a powerful word without intending to wield it to hurt me. Common sense always prevails. If you don't mean to cause me offence, I won't take offence, regardless of the words you are using around me and to describe me.