No, it can't. For a start, it doesn't mean 'happy' in all its senses - a better synonym would be 'very pleased [about something]'. For example, you might hear someone say 'I was a really chuffed when I heard I'd won the competition'. In other words, it's a reaction or response to some news or an event. But beware: It's very informal British English and it's only used by a minority of people. I've heard people say this, but I would never use it myself, nor would any of my family or friends. Should you use it? Definitely not. Most English-speaking people in the world would not understand you, and even those who did would be very confused to hear a Chinese person using such an unlikely word. Instead, you should use a standard word. 'Delighted' would work fine.

I would say...yes and no. I think this is fine: "I'm really happy about my new job" can be "I'm really chuffed about my new job". This would be a little weird: "Eating this cake makes me really happy" cannot be "Eating this cake makes me really chuffed".

'Chuffed' is more for things which you are extra happy about, like you had a stroke of luck. So as Andrew mentioned, you can be 'chuffed' about a new job, because you might feel lucky to have gotten it. But doing an everyday thing which makes you happy is not something you would say makes you feel chuffed.

Just for the record and to try to add to some of the examples the others have given. I would use it in the situation where the result of a situation is better than you had expected. Quite often if someone has complemented you or you are rewarded/ get recognition for doing something well, being right about something, or something you have done has turned out well. Again, especially if you didn't really expect to be recognised for your efforts. I think a person who is a long way superior to the others in a competition doesn't feel 'chuffed'. So this would fit in with what others have already mentioned about getting a job, winning a competition. I think it is quite a nice expression to use as it shows bit of modesty.