I know we use " Tom and I ", but can we use "Me and Tom" as well? some people say it is wrong or that it is not american english.

Also can someone tell me the basic difference in writing styles of British, American and Australian English?

Yes, you can use "me and Tom" when you would use "me" if it were just you. If you remove "and Tom" then use whichever pronoun you would nromally use (I for the subject and me for the object).

Please make sure you copy both me and Tom on your responses.

It came as a surprise to me and Tom.

Do NOT use "Me and Tom" as the subject of your sentence. That is, do NOT write "Me and Tom hope you will be able to come."