the first 了 is used to indicate the complete of an action. The second 了 is used as an interjection to emphasize. so, the first is necessary. so, your second sentence is perfectly okay.

I can’t understand your question

The sentence translates into "I had already ...." the 2nd 了is for emphasis. Omitting it, translates to "I had bought..."

已经 usually goes hand in hand with "了” ，meaning something has been done. I think "我已经买衬衫了" or "我已经买了衬衫” is grammatically correct. The trick here is that if you want to be specific about what has been finished, there will be two "了". 我吃了饭了=我吃了；我看了这本书了=我看了

我已经买了衬衫了means I have bought shirts yet