新たな and 新しい both mean the same, "new", however 新たな is more formal. About はず, it's like "ought, should, must, can, be sopposed to...depending on the sentence/situation. Here are examples: いま、9時半です。授業はもう始まっているはずです。(いま、くじはんです。じゅぎょうもうはじまっているはずです。) The class ought to have started / is supposed to have started / should have started / must have started. 彼はアメリカ人のはずです。(かれはアメリカじんのはずです。) He must be / can be American. 手紙は今日届くはずでした。(てがみはきょうとどくはずでした。) The letter was supposed to arrive today (but not yet). I hope this was helpful.

新しい & 新た are the same. But I think 新た has somewhat more formal nuance.