1. 好像才开始放假,马上又开学了。
It doesn't change much when omitting "开始" here, the word "开始" is actually for emphasizing "starting" and you haven't done anything...
But omitting 马上 will change the tense. "又开学了" means the semester "already started". To correct it, we need to add “要”, as it imply future tense.
For example,
又要开学了 .
Actually, 马上又要开学了 have the same meaning, here "马上" emphasize "soon it will "
Skip both without changing meaning:
好像才放假,又要开学了。
2. we may put 才 before time or a verb.
才两点呢 ( it's just 2 o'clock!)
又两点了 (it's 2 o'clock again !)
才买
又买
But for your sentence, it's more natural to keep the original order。
我姐姐昨天才买一双鞋子,今天又买一双!
4. Sure, it's grammatically correct to skip "早上" or "晚上". It just skipped some information.
Skipping the noun at the end is OK ( and we always do :) ), but usually add "了".
...., 晚上又去看了。