One and two you cannot remove it but for three and four you can. This is because the word "being" before another word means that the word (in #1 this word would be "poor") describes the person ("he" in #1). You can reword it to have the same meaning in these examples:
Number 1 means: he is poor but he is also happy.
Number 2 means: he is fast but he is also terribly overweight.
However, #'s 3 & 4, you can remove being because "from" in #3 and "as" in #4 already link the adjectives ("accurate" in #3 and "vulnerable" in #4) to the subject ("the conclusions" in #3 and "her" in #4).
These ones, you cannot reword it into the format as before because there is only one adjective in each sentence.
Hope this helps.