Translation

علامہ اقبال کو مسلمانوں کی غربت اور پسماندگی کا بڑا دکھ تھا۔ آپ نے ہندوستان کے مسلمانوں کو آزادی حاصل کرنے پر اکسایا ۔ آپ نے اپنے اشعار سے قوم میں اک نئی روح پھونکی ۔ آپکی شاعری امید سے بھری ہوئی ہے۔ آپ نے ہندوستان کے مسلمانوں کے لیے ایک الگ وطن کے قیام کا مطالبہ کیا ۔ آپ اسلام اور پیغبر اسلام کے سچے عاشق تھے۔ آپ ایک عظیم فلسفی بھی تھے ۔ ہمیں اپنے قومی شاعر پہ ناز ہے۔

Allama Iqbal was greatly grieved at the Muslims' poverty and backwardness. He urged the Indian Muslims to get freedom. He infused a new spirit/he put a new life into the nation through his poetry. His poetry is full of hope/optimism. He demanded a separate country for the Muslims of India. He was a true lover of Islam and the prophet of Islam. He was also a great philosopher. We are proud of our national poet.

نیچے دی گئی لائن کا اردو میں ترجمہ کریں:

سنی سنائی باتوں پہ یقین نہیں کرنا چاہیے-

First of all, I would like to mention that "hearsay" is uncountable in English, so we don't pluralize it or we don't put ‘a’ before it.
Don't believe hearsay.✓
Don't believe hearsays۔✖️
Rumor is AmE and rumour is BrE, so both spellings are correctly correct. 

• Don't believe rumours/rumors.✓
• Don't believe gossip.✓
• Don't believe in gossip.✖️
• Don't believe gossips.✖️

"Gossip" is uncountable when it means talk or stories about other peopleʼs private lives, that may be unkind or not true. 

Do you want me to tell when it's countable? Okay! It's countable when it means a conversation about other people and their private lives.
I love a good gossip.✓
Remember, it's countable but still singular.
Don't put your trust in gossip.✓
Another important point: "To believe in" means to have faith in. "To believe" (without "in") means to regard something as true:
I believe everything he says.✓
I believe in Allah.✓

میں چاہوں گی کہ آپ اپنے وعدے پر رہو-

I want you to stick to your word.✓
I want you to keep up your promise.✓
I want you to honor your promise.✓
I want you to fulfill your promise.✓
I want you to hold yourself to your promise.✓
All of these are correct; however, the best ones which suit your lines in the post are the first three. Other two are correct as well and fulfill the requirement of the context in the post. But Pakistani always want the word for/by word translation.