Check out, please. 请帮忙检查/改正我的作业

1) 我请一位教师告诉古代史我。(I asked my teacher to tell me about ancient history)

    我请我的老师告诉我有关古代历史的知识。
2) 他告诉我手机他。(He told me to call him)

    他让我给他打电话。

    ask sb to do sth--让 sb+do

    A call B--A 给 B 打电话
3) 教练叫给你们打蓝球。(Coach ordered us to play basketball)

    教练让我们打篮球。
4) 他让我说对你去睡觉。(He asked me to say you go to bed)

    他让我叫你去睡觉。
5) 爸爸叫我吻她。(Father told me to kiss her)

6) 我做过作业。(I've done my homework)

    我做完作业了。

Verb + 过 = Past Perfect?

yes,but it emphasizes an experience,like I have been to Beijing--我去过北京。I have watched this moviie--我看过这个电影.Your sentense:I've done my homework.In this sentense it is just one thing you have done so we use v 完 n .
What difference between 请, 叫 and 让. These words are synonims, right?

请--in a polite way.

让 is more often used in our daily life.

让 is more polite than 叫。

Check out, please. 请帮忙检查/改正我的作业

1) 我请一位教师告诉古代史我。(I asked my teacher to tell me about ancient history)

我问老師告诉我古代的歷史
2) 他告诉我手机他。(He told me to call him) 他叫我打[个电話]給他
3) 教练叫给你们打蓝球。(Coach ordered us to play basketball) 教练長命令我们打篮球
4) 他让我说对你去睡觉。(He asked me to say to you, "go to bed") 他叫我跟你說, "去睡覺"
5) 爸爸叫我去吻她。(Father told me to kiss her)
6) 我做过作业。(I've done my homework) 我/做完/完成/功課了

Verb + 过 = Past Perfect? yes but not always. Depends on context.  吃过了 = [I] Have/had eaten. You need more modifiers to specifically indicate Past Perfect. 昨天已經吃过了 = yesterday I had already eaten. Therefore in this sense, Chinese grammar is much easier; just use more time indicators rather than changing tense forms.

What difference between 请, 叫 and 让. These words are synonyms, right?

No.

请 = please [to make the sentence more polite] 请叫 = please call

叫 = to shout / to call / to order / to ask / to be called / by (indicates agent in the passive mood)

让  = to let.  eg "Let me call" 让我去叫

1) 我请一位教师告诉古代史我。(I asked my teacher to tell me about ancient history)
我请一位老师给我讲讲古代历史。

Check out, please. 请帮忙检查/改正我的作业

1) 我请一位教师告诉古代史我。(I asked my teacher to tell me about ancient history)

    我请我的老师告诉我古代史（历史）
2) 他告诉我手机他。(He told me to call him)

    他叫我打电话给他。/他叫我打他手机（cell phone）。
3) 教练叫给你们打蓝球。(Coach ordered us to play basketball)

    us=我们
4) 他让我说对你去睡觉。(He asked me to say you go to bed)

    他叫我说，你去睡觉
5) 爸爸叫我吻她。(Father told me to kiss her)

    
6) 我做过作业。(I've done my homework)

Verb + 过 = Past Perfect?

Actually it also refer to past tents. It depands. 

What difference between 请, 叫 and 让. These words are synonims, right?

请=please， in a polite way

in my opinion, 叫  和  让 is no big difference when using.