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
I've found a few sources that say to use a comma before the quotations, but they all seem to have different standards.
This one just says to use them all the time.
- Use a comma to shift between the main discourse and a quotation.
This source has a rule for short quotations.
Rule 16
Use commas to introduce or interrupt direct quotations shorter than three lines.
Examples:
He actually said, "I do not care."
"Why," I asked, "do you always forget to do it?"
And this last source has different rules for different situations.
First, use a comma to separate quoted material from the rest of the sentence that explains or introduces the quotation. If you are splitting the quotation with attributing the person, you will need two commas.
"The question is," said Alice, "whether you can make words mean so many things."
But if you are introducing the quote with the word that or the quote is only a small segment of the sentence, do not use a comma.
Peter Coveney writes that "[t]he purpose and strength of […]."
We often say "Sorry" when we don't really mean it.
The last thing explains when to use a colon instead of a comma.
Use a colon to set off explanatory or introductory language from a quoted element that is either very formal or long (especially if it's longer than one sentence):
Peter Coveney had this to say about the nineteenth-century's use of children in fiction: "The purpose and strength of […]."
Best Answer
In that type of construction, use "As". You will often hear "like" used in that and similar constructions, but it is very informal.