Is it permissible to say, "My Lord, grant me my requests," or "Give me my requests," to shorten the prayer time between the Adhan and the Iqamah?

There is no objection to supplicating with what you mentioned. The best supplications are the comprehensive ones that the Prophet (peace be upon him) used to make and teach his Companions, such as: "O Allah, grant us good in this world and good in the Hereafter and protect us from the torment of the Fire." And the supplication: "O Allah, I ask You for all good, now and in the future, what I know of it and what I do not know. And I seek refuge in You from all evil, now and in the future, what I know of it and what I do not know." And the supplication: "O Allah, rectify my religion for me, which is the safeguard of my affairs; and rectify my worldly life for me, in which is my livelihood; and rectify my Hereafter for me, to which is my return." And the supplication: "O Allah, I seek refuge in You from inability, laziness, cowardice, miserliness, old age, and the torment of the grave." And the supplication: "O Allah, forgive me, have mercy on me, grant me well-being, and provide for me." And the supplication: "O Allah, I have wronged myself greatly, and no one forgives sins except You, so grant me forgiveness from You; indeed, You are the Forgiving, the Merciful." And the supplication: "O Lord, help me and do not help against me; grant me victory and do not grant victory over me; plan for me and do not plan against me; guide me and make guidance easy for me; and grant me victory over those who oppress me."

Praising Allah the Almighty is a form of supplication, and frequent prayers upon the Prophet (peace be upon him) are sufficient for worries and a means of forgiveness for sins.