How do you deal with multilingualism here on AG
If someone, say, posts in the Newcomers Forum in Spanish, "Hello people, I love this site. Who else lives in Mexico or speaks Spanish? Speak it here!" I might respond on their profile with, "Hola y bienvenidos a Armor Games. De acuerdo con las reglas, los foros son Inglés solamente, así que por favor traducir el texto antes de su publicación." It seems far more accomodating and effective than only having a mod lock/delete their thread 'without justification' in their eyes.
If you mean when a person attempts to use English and it's a mess, I tend to ignore the errors and seek out their message. Usually it's somewhat readable with the context of either what they quoted or the topic of the thread.
or on the internet in general?
Google translate to get an idea of what they're saying.
Which problems does it cause,
Often a few communication roadblocks, especially when one side uses extremely poor grammar, uses idioms/puns/slang/sarcasm, misspells words, or uses 1337/txt spk.
are there any pros to either side as well?
Not really.
Does your cultural background have any influence on how you communicate with others
Of course it does. Heck, there are many regional differences within English. For example, to "table" something in the US means to "set it aside for now" while in the UK it means to "address it immediately or work on it".
or how you behave on the internet,
Of course. We've all got different backgrounds. We've all got some stereotypes of others in different regions and other nations (which don't necessarily need to be negative: Canadians are friendly, Swiss are efficient, Asians are leet, etc., but can still cause misconceptions when attributed to specific individuals) whether we admit it or not. Tolerance and acceptance move us forward.
And, well, what is the most annoying part of speaking to someone that is/isn't a native English speaker?
Usually talking past each other due to perhaps a mistranslated word. Or being left out of the conversation entirely, but that goes both ways.