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
The word as in your examples is a conjunction.
It is used as a conjunction to express similarity. You can think of it (this as) as like, where like can be used as a more informal version of as. You can write it in normal word order ("as her mother is" or "as I do"), but keep in mind that you could also find the inverted word order used in a very formal style.
Here are some related sub-entries from Practical English Usage by Michael Swan.
326 like and as: similarity, function
We can use like or as to say that things are similar. We can also use as to talk about function – the jobs that people or things do.
326.2 as (similarity): as I do
As is a conjunction. We use it before a clause, and before an expression beginning with a preposition.
as + clause
as + preposition phrase
Nobody knows her as I do.
We often drink tea with the meal, as they do in China.
[...]
326.4 inverted word order: as did all his family
In a very formal style, as is sometimes followed by auxiliary verb + subject (note the inverted word order – see 302).
She was a Catholic, as were most of her friends.
He believed, as did all his family, that the king was their supreme lord.
Here is the related part of the entry 302 mentioned above.
302 inversion (1): auxiliary verb before subject
302.4 after so, neither, nor
In 'short answers' and similar structures, these words are followed by auxiliary verb + subject.
I'm hungry. ~ So am I.
I don't like opera. ~ Neither/Nor do I.
302.5 after as, than and so
Inversion sometimes happens after as, than and so in a literary style.
She was very religious, as were most of her friends.
City dwellers have a higher death rate than do country people.
So ridiculous did she look that everybody burst out laughing.
Best Answer
"Due to the fact that" is an idiomatic expression often used to say "because", it does not refer to a specific fact.
"Due to that that" is not used and would probably not be understood.