Should every income have a corresponding asset?

For the first entry, yes, pretty much. If a company receives money from Income that is recorded as income and cash, cash being the asset account. If the company records the income as a recievable, meaning the customer hasn't paid yet, then the account would be an Accounts Receivable, also an asset account.

Though income is recorded on the Income Statement and not the Balance Sheet, the revenue from said income becomes an asset, adjusting entries later on the books will take care of anything else, such as merchandise, inventory, cost of goods sold (if a merchandising company) or other expenses related to said income.