Yes, a draught, or as written today, draft, is a written direction ordering the payment of money from an account. Checks are sometimes still referred to as drafts.

I am unable to answer the rest of your questions about drawing funds from others. It appears to be an alternative to using banks. As banks were less regulated then (remember one of Austen's brothers had a bank that went under), and probably not located in every town, I can imagine the wealthier people in a locality assisting with these types of transactions.