Does Core have reports that break down the Income, Consultant Fee and Reimbursable amounts for architects?  

Core offers many detailed and summarized reports to serve different requirements of different companies. As long as you set up your Chart of Accounts correctly and assign the income and expense accounts in the Global Settings, Projects or Lists > Activity/Expense Items screens, you can get the desired breakdown of your income and expenses in the financial reports. Example, the Profit and Loss reports in Core.