By far the most common expression for an announcement on Twitter is a 'tweet'.
A lot of television shows now have an interactive element where they ask the viewers to 'tweet' their questions or opinions to them.
With regards to your second question, I would say 'Google it' refers to the use of Google only. That's just my opinion however, I can imagine some people using the expression when they have used another search engine, but I think the majority of people would use Google.
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