Whoever made up Hungarian is a genius - what takes seven words to say in English takes only four in Hungarian!
I would say it depends on the context how it sounds better: with or without the indefinite article. E.g. when Mom instructs her sibling not to sit on the table, she might say "Ez asztal, nem szék." When Mom orders a chair from the net and a table is delivered instead, she might complain to the courier: "Ez egy asztal, nem egy szék." But these are nuances, you should be fine with either way.
It's omitted because it isn't necessary. The fact that it is a chair is not what is important, but rather, that it is a chair
why is "ez" translated with "this"? and why is it sometimes translated with "that"? it's so confusing