The correct pair among the options provided is:

Option 2: Battle of Wandiwash - French vs. East India Company

Reasoning:

1. Battle of Buxar: The Battle of Buxar took place in 1764 and involved the British East India Company against the combined forces of Mir Kasim, Shuja-ud-Daulah, and the Mughal Emperor Shah Alam II. Mir Jafar, who is mentioned in Option 1, was not involved in this battle. Therefore, Option 1 is incorrect.

2. Battle of Wandiwash: The Battle of Wandiwash, also known as the Battle of Vandavasi, occurred in 1760 during the Seven Years` War. It was fought between the French East India Company led by Comte de Lally and the British East India Company. The British, under the command of Sir Eyre Coote, emerged victorious in this battle. Hence, Option 2 is correctly matched.

3. Battle of Chilianwala: The Battle of Chilianwala took place in 1849 during the Second Anglo-Sikh War. It involved the British East India Company led by Sir Hugh Gough against the Sikh Empire under the command of Sher Singh. The reference to Dalhousie in Option 3 is incorrect, as he was the Governor-General of India during the time of this battle and not personally involved in the fighting. Therefore, Option 3 is incorrect.

4. Battle of Kharda: The Battle of Kharda occurred in 1795 between the Maratha Empire, led by Nana Phadnavis, and the Nizam of Hyderabad, supported by the British East India Company. The reference to the East India Company in Option 4 is correct, but the Nizam was the opposing force, not the East India Company itself. Hence, Option 4 is incorrect.

In summary, the correct pair is Option 2: Battle of Wandiwash - French vs. East India Company, as it accurately matches the opposing forces in that battle.