"The restaurant is open."
Translation:Restaurangen är öppen.
How exactly is "... har öppet" used? Essentially means "has 'openness' "?
That form only occurs before the noun: den öppna restaurangen but restaurangen är öppen. These two different positions for the adjective are called attributive and predicative. I posted a table here which shows all cases:
No, the past tense of öppna is öppnat.
öppen is an adjective and those don't have tenses. It's just that we have two different constructions to choose from so that you can say either that Restaurangen är öppen or Restaurangen har öppet. Both mean 'The restaurant is open'.
ha öppet only refers to things like 'have opening hours' so that for instance a window can only vara öppet, not ha öppet.