I am a bit confused about the use of 'that' and would

I don't think your question can easily be answered. It is a bit more complicated than that - it really depends if you have two independent clauses joined together with a "that" or if you have just a sentence with an additional noun that happens to use a "that". Also, more often than not, GMAT will test a VPR (Vague Pronoun Reference) error by throwing that or it, making the explanations even more confusing. Here is a primer on the use of that from the best source I have found so far - TOEFL Cliff's Book. Here is a quick summary of the rules:


1. Relative Clauses
A relative clause is used to form one sentence from two separate sentences. The relative pronoun replaces one of two identical noun phrases and relates the clauses to each other. The relative pronouns are:
- that
- which
- who
- whom
- whose

Remember that a sentence with relative clause can always be reduced to two sentences, so each clause must contain a verb

Restrictive and Nonrestrictive Relative Clauses (that)
A relative clause can be either restrictive or nonrestrictive. A restrictive clause is one that cannot be omitted from a sentence if the sentence is to keep its original meaning. A nonrestrictive clause contains additional information which is not required to give the meaning of the sentence. A nonrestrictive clause is set off from the other clause by commas and restrictive clause is not. Who, whom, and which can be used in a restrictive or nonrestrictive clauses. That can be used only in restrictive clauses. Normally, that is the preferred word to use in a restrictive clause, although which is acceptable. TOEFL does not test the use of which and that in restrictive clauses.

Examples of restrictive and nonrestrictive clauses:

Restrictive: Weeds that float to the surface should be removed before they decay.
(We are not speaking of all weeds, only those that float to the surface. Thus the sentence is restrictive; if "that float to the surface" were omitted, the sentence would have a different meaning.)

Nonrestrictive: My car, which is very large, uses too much gasoline
(The fact that my car is very large is additional information and not important to the rest of the sentence. Notice that it is not impossible to use the pronoun that in place of which in this sentence)

Examples of relative clauses:
Hurricanes that are born off the coast of Africa often prove to be the most deadly.
Film that has been exposed to X rays often produces poor photographs.

That - Other Uses
* Optional: the word that has several uses besides its use in relative clauses. One such use is as a conjunction. Sometimes when that is used as a conjunction it is optional, and sometimes it is obligatory. That is usually optional after the following verbs:
- Say
- Tell
- Think
- Believe

* Obligatory: That is usually obligatory after the following verbs when introducing another clause:
- Mention
- Declare
- Report
- State

* That Clauses: some clauses, generally introduced by noun phrases, also contain that. These clauses are reversible.
E.g. It is well known that many residents of third world countries are dying.
OR
That many residents of third world countries are dying is well known.

Subjunctive
The subjunctive in English is the simple form of the verb when used after certain verbs indicating that one person wants another person to do something. The word that must always appear in subjunctive sentences. If it is omited, most of the verbs are followed by the infinitive

We urge that you leave now.
We urge you to leave now.

The following verbs require that after them
Advise, ask, command, decree, demand, insist, move, order, prefer, propose, commend, request, require, stipulate, suggest, urge

The following structure applies:



Also, a similar rule applies after impresonal expressions with the same meaning as the above verbs. The adjectives that fit into this formula include the following:
Advised, important, mandatory, necessary, obligatory, proposed, recommended, required, suggested, urgent, impreative

The following structure applies:



E.g. It is necessary that he find the books.
Note that it is not finds


Source: TOEFL Cliff's Book.