I think you are confusing the Ethereum (Classic) blockchain and a general block explorer. The blockchain simply holds the data and state of everything that has ever happened. A block explorer is a visual representation of all of the data from the blockchain, as getting data directly from the blockchain for every query is highly inefficient.
how can I determine the amount of Smart Contracts that are hosted on
the ETC blockchain?
There are a few ways to do this, but will be difficult. The most efficient way would be to write a script that scans the blockchain for each instance of a contract creation.
You can use geth for this, but there are other options, if desired.
Also, of these smart contracts, how many of them have been verified?
"Verified" is strictly a statistic created by the hosts of block explorers. There is no such thing as a verified contract from the perspective of the blockchain. I am not sure there is a block explorer that does Verify contracts for ETC.