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: 
I know a German girl here in Florida, who obviously learned British English in school. But after living in Florida for over 10 years, she has an immaculate American accent. The problem is that she still uses "bloke" where Americans would use "guy," she says "you know 't I mean?" with a British intonation but American accent, she says "zed" instead of "zee" and says "shedule" instead of "skedule." You would think these innocuous little details that nobody really notices, but the fact is that a lot of people find her amusing and even laugh at her (even some of her friends.) I personally find the inconsistencies jarring - especially "bloke." Every time she says it, I feel like saying "don't say that!" hahaha
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.