Does Islam grant respect to women?

I would say that Islam grants even more respect to women than to men. According to one Hadith a man once came to the Prophet and asked him who rightfully deserved the best treatment from him.

“Your mother,” said the Prophet. “Who’s next?” asked the man. “Your mother.” “Who comes next?” the man asked again. The Prophet again replied, “Your mother.” “Who is after that?” insisted the man. “Your father,” said the Prophet.

Another example concerns Hajra, the Prophet Abraham’s wife. Hajj, regarded as the greatest form of worship in Islam, entails the performance of Sai, one of the main rites of the Hajj. This is accomplished by running back and forth seven times between Safa and Marwah, two hillocks near the Kaba. This running, enjoined upon every pilgrim, be they rich or poor, literate or illiterate, kings or commoners, is in imitation of the desperate quest of Hajra, Abraham’s wife, for water to quench the thirst of her crying infant, four thousand years ago. The performance of this rite is a lesson in struggling for the cause of God. It is of the utmost significance that this was an act performed by a woman. Perhaps there could be no better demonstration of a woman’s greatness than God’s command to all men, literally to follow in her footsteps.

We can see that the principle implied by the expression ‘ladies first’ in modern times had already been established in Islam at the very outset.