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Let me first offer some examples to clarify the distinction between "which" and "that".

Suppose the other room has some books, and I issue one of the following commands:

1. Get me the red book, which is mine.
First of all, note the comma after "book". The comma is an important separator on the GMAT. In this sentence, the comma tells us that what follows it is extra information about the book. So you should be able to remove "which is mine" from this sentence, and the command should still make perfect sense. This suggests that there is exactly one red book in the other room.
The command in this case translates to the following two sentences:
Get me the red book. The red book is mine.

2. Get me the red book, that is mine.
This sentence is incorrect. You cannot have the comma and "that" together like this. Note that "that" can start an independent sentence. So the above construction has two independent sentences without a conjunction to connect them. "That" is not allowed.

3. Get me the red book that is mine.
Note that there's no comma in this command. This means the "that" provides essential information to unambiguously identify the book I am interested in. The command suggests that there are possibly many red books in the other room but that only one of them belongs to me.

4. Get me the red book which is mine.
Again, there's no comma in this command. This command provides the same information that is provided in 3.
There is some confusion over whether you can use "which" without the comma to provide essential information. The GMAT official guide has a couple examples in which "which" is used without a comma. The official guide doesn't call out those sentences as incorrect, but it admits that the rule is controversial.

The bottom line is that, if you are forced to choose between 3 and 4, you should prefer 3. But I do not expect the GMAT to make you choose between the two without providing other markers to adequately separate the correct answer from the incorrect ones.

The scope of "which" is restricted. Typically, it can modify only the noun immediately preceding it. In some cases, it can modify a noun that's slightly far away. For example: He composed an album of songs, which is/are popular. In this construction, which can modify either album (in which case you need "is") or songs (in which case you need "are").

If you want to modify an entire clause/an action, you should use an "ing" modifier. Alternatively, you can use a word or a phrase that captures the essence of the action (sometimes called a "resumptive modifier" because it introduces some repetition to eliminate ambiguity).

In this question, both B and E are correct. The GMAT would not make you choose between them and would typically place other errors in the sentences. But if it makes you choose, you should choose "that", as there's no confusion about its usage in restrictive clauses that do not employ the comma.

Hope this helps.

--Prasad