Learn English – the same company as / the same company that – Which one is the best choice

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

You can't omit that when it's a subject, or your sentence becomes ungrammatical in standard English.

In this case, that is the subject of a that-relative clause:

There is an engine inside me
　　[ that keeps saying "someone must be at the top, why isn't that you?" ]_{that-relative clause}

In particular, it's the subject of the verb keeps. So you can't omit it, and there's no formal vs. informal distinction to be made.