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 looks weird to someone who is used to checking spelling, for example anyone who produces or checks professional copy. They're used to noticing single errors like typos, and common mis-uses like "flout" vs "flaunt" in everything they read, but they might well perceive what you're doing as a consistent mis-spelling.
* Individual words might look weird (but whatever spelling you use, *someone* who's used to the other will find it weird, so this may not be a problem).
* It's "wrong" in the sense that it's not correct British English and it's also not correct American English. But from what 200\_success says, almost any mixture is correct Canadian English, so you can plead that. Furthermore, only people very familiar with multiple versions of international English will be absolutely certain that what you're writing isn't conventional *anywhere*. So you can get away with being slightly "wrong".

I would say that in your own informal writing you can do as you please. Few readers will both notice *and* care, so even if it looks a little odd it won't seem definitively incorrect.

If you're writing formal English then you should pick a spelling convention and stick to it. It's part of a style guide. This goes beyond even choosing British vs. American: for example there are plenty of words which in Britain can correctly be spelled either *-ize* or *-ise*, but generally you should spell all of them the same in a given text, and certainly you should spell each one consistently. This is something that spelling/grammar checkers in my experience don't pick up on: "you've used 'spelled' in this paragraph and 'spelt' in that one, would you like to be consistent?".

So in practice, many native speakers make mistakes along the same lines, especially a Brit trying to write American English for an international audience. Therefore, although it's *better* to stick to a common set of spellings, it won't stand out all that badly if you don't.
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.