In sentence 1, that acts as a relativizer (relative pronoun). It may be dropped (in any register) because it acts as the direct object of the verb in the relative clause. It could not be dropped in formal English (though it often is in informal spoken English) if it were the subject of the verb in the relative clause:
✲ It's the same girl Ø took our family photo.
In the remaining sentences, that acts as a subordinator (subordinating conjunction). In sentences 2, 3 and 4, that may be dropped because the subordinate clause which it heads is the direct object of the verb in the main clause and is in its ordinary position immediately after that verb.
If that played another role, such as subject, or if the subordinate clause were displaced to another position, that could not be dropped, because it would not be clear that it is in fact a subordinate clause:
✲ Ø he's protected by his family is understood by Alex ... The clause falls at the beginning of the sentence, before the verb is, because it has become the subject. That cannot be dropped.
✲ I came to know some eight or ten days after I got the report Ø you got stuck in traffic. ... Here the subordinate clause has been separated from its governing verb by a fairly long ('heavy') adverbial phrase . You could probably get away with dropping the that in speech, but it cannot be dropped in formal writing.
In sentence 5, and in these rewrites of sentences 4 and 5, the situation is a little different: These subordinate clauses are predicative complements of BE, and in speech that may be dropped even if the clause is moved to the front. In writing it's permitted, but not advisable; you really want to give the reader as many clues to your structure as possible:
? What many people are saying is Ø they saw a ghost.
? Ø I need help from you is the reason I'm helping you.
In other uses, as a demonstrative adjective or a demonstrative pronoun, that may not be dropped.
I want that puppy. but not ✲ I want puppy.
John took that from Shakespeare. but not ✲ John took from Shakespeare.
✲ marks an utterance as unacceptable
? marks an utterance as possibly unacceptable
Ø marks the place where that is omitted
It is not a “standard phrase” but a quotation from Edmund Burke. (I have not read enough Burke to identify the exact source from memory.) We do not misquote quotations because we do not like their style. Burke’s is the high literary style of the eighteenth century.
Your rewriting is not grammatical. Here is a grammatical version:
The triumph of evil requires that good men do nothing.
However, that rendering does not express the original meaning.
The triumph of evil requires only that good men do nothing
Your analysis of the grammar of the original is wrong. “For good men to do nothing” is the complement to “necessary.” The grammar is hard to decipher because Burke plays with word order, ellipsis, and the wide field of usage of “for” to render an implausibly optimistic thought memorable and plausible. Here is a slightly revised version that makes the grammar and logic easier to analyze.
[In order] for evil to triumph, it is [both sufficient and] necessary for good men to do nothing.
Best Answer
Neither is very common, except in speeches given by people learning English.
"There are four people in my family." is a simple and correct expression.
It is probably more common, if you are asked to talk about your family, to describe them:
Some families are simple
You don't need to say "five people" because we can count.
On the other hand, some are more complex
Families are complex: consider the example. How many people are in that family?
It would normally be "me" because that word is not the subject of a clause. In this respect, English is different from some other languages (in particular Latin and some Latin derived languages). There is some flexibility on this. "It is me." is normal and correct. "It is I" is also correct, but less used. In a long phrase "My mother, my father, my sister and I" could also be correct, but unnusual. English tends to reserve "I" for first-person singular subjects.