How is inheritance classified in Adam Smiths profit/wage/rent classification? It seems to me that inheritance can't be classified as profit or wage. So does it fulfill the definition of rent, or are there some economic activities that don't fit into this classification?
Profit, wage and rent are rewards for inputs used in production, they constitute a decomposition of Income (or the Income-decomposition of output produced)

Inheritance is a transfer, it does not involve production. So it is invalid to try to fit it into the mentioned classification.

Consider now any gift: say I work, earn a wage in the form of consumer goods, and give them to you. The consumer goods I got as a wage is a reward for my labor. Giving them to you is a transfer of already produced output. 

In inheritance, this output has already taken the form of assets rather than consumer goods.