Hi Hive account@noahcramos, I'm not studying blockchain tech like you do, but will try my best to address your questions. Do correct me if I am wrong!

when transactions occur through payment channels on the lightning network they are off-chain correct?

Yes, it is based on an additional layer of network on top of the BItcoin blockchain.

For instance if a payment channel was opened between a contractor and the person funding the project, those small transactions such as hardware store trips and other costs would be conducted through this channel off the blockchain?

Yes, a payment channel is opened between two wallets, that are already registered on the blockchain. As many transactions can occur between these two wallets through the channel, as long as the balance is kept in check. When the channel is completed and closed, the final balance is recorded.

If i understand this correctly then the first and last transactions of the channel are the only ones recorded?

Yes, secured channels between the people you are connected with, is sufficient to perform micro payments. Hence a path of shortest resistance is used. When used together with the main blockchain, it verifies the validity of such channels.