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 phrase "get to do something" implies to have the opportunity to do something. So the meaning of the first sentence is "I love to have the opportunity to see her first thing in the morning".
The phrases "get to do something and get to doing something" aren't interchangeable to convey the same meaning. The latter phrase means "to start doing something such as I got to talking to her. I got to thinking that it was his fault.
Both the sentences are grammatically correct, but they convey different senses.
Best Answer
As you say, "started" is a past perfect participle which effectively is an adjective,
It seems to me you can use the "get + past perfect participle" with any verb where the past perfect participle is a similarly idiomatic adjective.
Or, more colloquially:
The past perfect participle of "write" is "written", and it's perfectly fine to say something like:
in the same way you might say:
"Written" here is an adjective that describes the intended status of the report.
However you do have to be careful since the meaning of "written" can vary depending on context. A "written report" is usually considered to be one that has been written by hand not one that is completely finished. As usual, you have to know what is idiomatic.