The party registration isn't counted as a vote for that party - when you show up on election day you can vote for whoever you want.

What it affects is which parties' primaries you can vote in.

The primaries aren't official US elections. They're unofficial elections set up by each political party (which are private, non-government entities).

You can vote however you want in official elections.

I would be careful about asserting that. In PA there can be referendums on the primary ballots. And the results can be binding (sore loser laws, etc.). They're pretty official looking around here.