Should 'leopard' be capitalized?

No, the word "leopard" should not be capitalised because it is a common noun but there are some exceptions which I discuss below when the word becames part of an individual cat's name. 

None of the wild cat species should be capitalised including the lion and tiger. It's interesting, however, that you still see the names of wild cat species sometimes capitalised. 

I believe that there was a convention perhaps a hundred years ago when these nouns were capitalized. Grammar, after all, is an artificial convention. It is not an absolute set of rules set in stone. 

Humans decide what is and what isn't acceptable in terms of grammar and the current thinking is that the word "leopard" should not be capitalised and neither should any other name of the wild cat species. 

Sometimes, however, you will see a lion that has been named because they are famous i.e. 'Cecil the Lion'. Clearly, the whole name should be capitalised as all names are. The whole name is 'Cecil the Lion'. The intervening 'the' should stay in lowercase.

Sometimes man-eating leopards have been named such as the the Leopard of Rudraprayag; a leopard reputed to have killed over 125 people. It was eventually killed by hunter and author Jim Corbett who has a tiger reserve named after him in the north of India.

The cat has been given a name and the full name is as stated. In this case the word 'leopard' is capitalized as it is part of a proper name (proper noun). Once again the intervening 'of' is in lowercase.

Amur leopard. Photo in the public domain.

And when there is a prefix which tells you where the cat is from, the pre-fix should be capitalized as is the case for: Amur leopard and Siberian tiger for instance.