For me, there is a difference between these. I use "actually" when I am about to contradict something that someone else just said, and I use "as a matter of fact" when I am about to reinforce something that I just said. Actually You can use both for the same purpose :-)

Well that depends on what you are writing. 'As a matter of fact' is a more formal style of writing and would seem more professional, certainly in a language exam it would earn you extra points for fluency. However 'actually' is not an informal word, it can be used perfectly acceptably in writing. Actually I'd say the that it is important to mix interchangeable words and phrases so that your writing doesn't sound repetitive. I hope this helps.

Of course it can, why not? The thing is your style may vary depending on the kind of writing, but words such as "actually" are not prohibited in any case, and will not be, no matter what some people say. Actually, replacing everything simple with longer, "smart" expressions, writing long, convoluted sentences etc. is a sign of bad style. There are specific guidelines in certain fields, however, depending on what you do (academic writing, journalism, technical documentation, legal documents, bureaucratic nonsense etc.)

Actually yes. Why not?

Yes.