Can some point me to all various usage of words - that and

I'm also not sure whether there's a name for #4, although it's likely one exists. The "that" serves to create a noun phrase, which itself is the object of the main verb. In other words, "I believe something" and the something is "everyone should have a puppy." You'll know you're dealing with an example of this when "that" follows a verb immediately and another subject/verb pair follow.

I'll throw in one more:

#5 Subjunctive mood: Our boss suggested that Larry attend the meeting.

This usage is similar to #4--"that" follows a verb immediately, and "Our boss suggested something."

The difference is that "Larry attend" is not a normal subject/verb pair (note that, on its own, we'd say "Larry attends a meeting." The verb in this "that" clause is hypothetical (our boss just wants Larry to attend, but he hasn't yet, and in fact may not), so it gets a special form: the infinitive (to attend) minus the "to."

For #5, the GMAT requires the "that" even though many people omit it in speech.
For #4, the GMAT seems to prefer to include the "that," also.