Question: I normally write using the *American* English forms, but for some particular words, I tend to naturally write it in the *British* manner (like with the word *favourite* rather than *favorite*).

Is it wrong to mix both styles? If not \*wrong\*, does it looks weird?

Answer: 
It must be noted, to add to the other answers, that some official exams will penalise you for mixing. For example, Cambridge (British English) will probably tolerate American English if you are consistent, but certainly penalise you if you mix them. I have also known of professors in the UK that consider American-spelled words an orthographic mistake, and therefore lower your grade for it (and being harsh!).
I would suggest that there shouldn't be any differences in the patterns you use as patterns are an important part of understanding what people are saying.
For example, don't say "my favourite color", or "my favourite colour is red and my favorite shape is a circle".

As a person who is very international, I have nothing against using words used in different areas(garbage/rubbish, etc.), and can accept that it may seem weird to use say Australian and American words mixed within British English, but only see it as an adaption and a sign of the diversity of the person's language and experience, and not something that is wrong.

Language is about 2 things, expressing yourself and communicating.
If something that sounds awkward does one of the above better than the standard, then I can not see what is wrong with it.
Language adapts, you just have to look at English 50, 100 and 150 years ago.