Please cut = أرجو أن تقطع / من فضلك اِقْطَعْ
?

@odexed sure can, very correct and almost same meaning.
I think please is a little polite.

Excuse me maybe more and more polite. And it's like (من فضلك)
Correct me please.

Sometime I feel أرجو is better if you're asking someone to do something which he will do later, not now. (This is my personal feeling) (Arabic language is really very deep, and even for us Arabs, you cannot easily find someone who is deeply educated the Arabic, sometimes we are confused which is better in each situation).

We use it also if you wish something, like: أرجو أن يكون متعاونا.
If you're going to meet someone, and you're wishing that he will be helpful.

I don't have a clear reference for that, but that's what I learn from all of my readings in Arabic فصحى books.

Can you please add two another translations that start with
أرجو منك أن تقطع / من فضلك اِقْطَعْ
?

Thanks for your very detailed explanation, I appreciate it. I think that this sentence has a little different meaning, it says "I want to cut...", it's different from what the English says. That's why I left my first comment. It's better to correct it than to add another translations.

I asked you to add the other 2 sentences because the 3 are correct and will give almost the same meaning and almost the same politeness level.

That's what I really like about tatoeba, translate the sentence to give the same native meaning (giving the sense of the original), not a literal translation.

I didn't say they are incorrect, but "Please cut the rope" is different from "I want to cut the rope", isn't it? If two translations don't match, we should either change one of them or unlink them.

hmm, they are different.
But I'm talking about the same sense!
أنا أريد أن أقطع
is different than
اقطع تلك الخشبة

I have a feeling that
1- (أنا أريد أن أقطع) gives the same sense of "Please cut" | "I want to cut ....".
2- (من فضلك اقطع) gives the same sense of "Excuse me, cut this ....".
3- (اقطع) gives the same sense of "Cut this .....".
4- (أرجو منك) | (أريد منك) gives the same sense of "I want from you to ...."
5- (أتمنى أن تقطع) = "I hope that you cut ...."
6- (هل تستطيع أن تقطع) = "Can you cut ...."
7- (هل يمكنك أن تقطع) = "Are you able to cut ..."

I would like to invite someone who is more native than me in English with the same Arabic level of me to discuss that together.

I like it. It's really challenging to reach a point were we give the most accurate translation.

I have another question here which I would like to mention some native English person here to answer.
@AlanF_US and @Objectivesea may check that

Which is more polite "Please," or "Excuse me,", if you want someone to do something for you?
Like if I say.

1. Please give me this pen!
2. Excuse me, give me this pen!

Or is it wrong to use excuse me in such a situation?

> "Please cut" | "I want to cut ....".

Perhaps my Arabic level is low, but I can't agree on this one. "Please cut" sounds more like you want someone to cut something for you while "I want to cut" implies that you are going to cut it yourself.
I think @Amastan could help us here.

I think some of these might be more naturally expressed in English than in your suggested translations..

4- (أرجو منك) | (أريد منك) gives the same sense of "I want from you to ...."
5- (أتمنى أن تقطع) = "I hope that you cut ...."
6- (هل تستطيع أن تقطع) = "Can you cut ...."
My suggestions:
4. I’d like you to....
5. May I ask you to cut....
6. Could you cut....

"Excuse me" has a few uses in English, but a very common meaning is a slightly interruptive one. If I walk up to a clerk in the hardware and lumber store and want to get his or her attention, I might very well say: "Excuse me." He or she will make eye contact and acknowledge my presence. Only then would I say: "Do you have tongue-and-groove flooring?" or: "Could you tell me where I can buy paint and paintbrushes?" or: "Do you cut lumber to specific sizes?" Once you're at the right place and have selected the wood that you want cut, there's no need at this point to say "Excuse me" a second time, but a "please" is certainly welcome: "Please cut this piece of wood...." (as above). Or you can just say "I'd like you to cut...." or "Could you cut...?" or "Could you please cut...?"

Finally, your last statements both look odd, but the first one would be okay with a period.
1. Please give me this pen!
2. Excuse me, give me this pen!

The exclamation mark is definitely not needed in these sentences. It is reserved for a few special uses — for example, to indicate a sound effect, to indicate strong emotion, or to indicate a direct command. As soon as you make it polite with a "Please" or a "Could you please", the exclamation mark becomes contraindicated.

I cannot conceive of any situation in which a native English speaker would utter the second sentence above.

By the way, I think it should be إلى instead of إلي

@odexed, I changed the sentence to من فضلك اقطع

You guys are great.
I'm now more convinced to change it.

The most important thing I learnt today is:
"Please" vs "Excuse me", thanks @Obecjtivesea

@odexed, woow perfect. ى yes, that absolutely right. Appreciate your comments bro.