I have to confess that I missed a few episodes in the 2nd season, as I'm a new fan of the show. I must have missed the one where Henry became a partner in the store.

Two questions: 1) Why did Jack sell him half of the store? and 2) how did Henry come up with the money?

He didn't sell it to Henry, he gave it to him. He felt that Henry had proven himself to be a capable business partner, and gave him half the store, making him a full partner in the business. Henry didn't have to come up with any money, because nothing was bought or sold.

Hope this helps.