我的中文课作业。

Hi! I'm studying Chinese at the university. Our last lesson was about "Things in a house". And while I was doing my homework (sentence translation of sentences), I realized I have some questions and haven't somebody [there is] no one to help me. So, they are there it is.

Is it right to say:

1) 妈妈让我帮助她整理客厅。。。or 妈妈让我帮助她在客厅整理。。。The first sentence is grammatically OK if your ellipsis is replaced by a period.  There should be nothing more to follow in that sentence.  The second sentence is fine if your ellipsis denotes something more to come afterwards such as "一下 [子]" (a little bit)

2) 客厅在对面。。。 or 对面是客厅。。。( Сan you give me some examples whith "对面"?)

Both are correct.  The first one translates into "The living room is on the opposite side facing..." and the second one is "... facing opposite is the living room".  The meanings are different.

可以跟我对面说嗎? Can you talk to me face to face?

对面鄰居姓李  The neighbour directly across is surnamed Lee

3) 下课以后你去哪儿？我去学院对面的花园。(I don't know the second sentence is correct or not)

You got it.  No problem at all.  You are going to the garden that is across the academy.

1) 妈妈让我帮助她整理客厅。。。or 妈妈让我帮助她在客厅整理。。。 
First sentence is correct. However, it would be more natural to say  妈妈让我帮她整理客厅
Second sentence, 整理 is usually used to mean "tidy" so you have to say 整理 xxx. It sounds strange because this sentence means "Mummy let me help her tidy in the living room." If you want to keep the sentence like this, you may add 一下 at the end. So 妈妈让我帮她在客厅整理一下 "Mummy let me help her tidy a little in the living room" makes more sense.

2) 客厅在对面。。。 or 对面是客厅。。
Both are correct.
Examples:
我在你家对面 – "I'm opposite your house"
对面有一座房子 – There's a house opposite"
你能到对面的房子去拜访吗？– "Can you go to the house opposite to visit?"

3) 下课以后你去哪儿？我去学院对面的花园
This is correct. In case you're unsure, the second sentence means "I'm going to the garden opposite the academy"