Short answers, pulled from the story.
Ottoman Sultan Selim I sent two Turkish engineers named Ustad Ali Quli and Mustafa Rumi to Babur's court in 1512 carrying matchlocks and cannons for use in open field battles. This tactical innovation gave Babur a decisive advantage over his rivals and became the foundation of Mughal military dominance in India.
The Mughal Empire lost control of Kandahar to the Persians in 1649 when they attacked and conquered the city after Shah Jahan had previously retained it in 1638. Aurangzeb made three attempts to recapture the city but the Persian resistance and harsh regional conditions made all efforts futile by 1668.
Mughal Emperor Akbar appointed Mir Haj as an official in 1577 and granted him 500,000 rupees and 10,000 Khil'lat to serve as a representative for the Shariff of Mecca. Akbar's wife Mariam-uz-Zamani also built a massive trading ship named Rahimi that traveled from Surat to Jeddah to carry merchandise and pilgrims.
English envoy Thomas Roe fell from favor at the court of Emperor Jahangir because the Emperor had a strong prohibition of wine in his court despite Roe initially gaining favor with gifts of red wine. Roe resided at Agra for three years and was allegedly Jahangir's drinking partner before the Emperor's prohibition caused Roe to end his political career in the Mughal court.
The Balkh campaign initiated by Shah Jahan in 1647 resulted in Prince Aurangzeb routing the Uzbegs under Abdul Aziz near Balkh but the Mughals were forced to leave the same year due to harsh winter and shortage of supplies. The campaign failed to secure the Mughal homeland of Samarqand and Fargana as the Oxus river was hardly defensible and no serious attempt was made for the former.
The Ethiopian Emperor Fasilides dispatched an embassy to India in 1664 to congratulate Aurangzeb upon his accession to the throne of the Mughal Empire. This diplomatic exchange marked a rare connection between the two distant powers and occurred alongside Mughal relationships with the Sultanate of Maldives and the Aceh Sultanate.