Like a thing other forums have where you can contact other user - yes I think so. also on facebook messanger and things like that.

Now text (on mobile) - is in theory also "wiadomość", but only people who translate mobile interface use it. normal people say SMS (esemes)

Not quite, it is awkward.

• If you want to say a neutral phrase, the word order should be "Adam wysłał ci wiadomość."
• But if you want to stress to whom the invitation was sent, you should use the long form of pronoun: Adam tobie wysłał wiadomość".