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 think most people would recommend you stick to one style or the other. Why? Well, it doesn't matter too much, but if you mix styles the reader might notice! And that's bad—if they're noticing stuff like that, then they're paying attention to *how* you're writing rather than *what* you're writing. 

In other words, you're distracting the reader. You probably don't want that. You want them to focus on what you have to say, right? So unless you have a specific reason to mix styles, just stick to whichever style seems appropriate.
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.