A church or any other public charity can form a subsidiary to operate a business for its own benefit.  But the business may not necessarily be a 501(c)(3) charitable organization.

If your housing corporation is going to buy houses and flip them to young professionals or others who pay market rates, you probably don't qualify for 501(c)(3) status.  To obtain charitable status you probably need to provide housing at below market rates or below actual cost to low-income people.  The economics of a 501(c)(3) housing corporation do not normally provide profitable margins to split with others. 

You can always run a housing corporation as a for-profit business and dividend the profits, after payment of income tax, to a controlling charity.  But I generally don’t recommend that a charity undertake a business activity that is not closely related to its exempt purpose.  It is hard enough for for-profit entrepreneurs to make a business profitable.  It is a lot harder for those whose experience has been in the nonprofit sector to run a business that turns a significant profit.

For more information on church management and administration consider this text;

Smart Church Management: A Quality Guide to Church Administration
By Patricia S. Lotich

Church leaders understand that managing the day-to-day operations of a church can be challenging because of limited resources, managing volunteer labor and supporting the needs of the congregation. Smart Church Management 2nd Edition is a guide to managing the resources of a church which is people, time and money. This book provides tools and examples for decision making and problem solving for church administration that is easy to understand and more importantly, quick to implement! This book explains how to create a strategic plan and takes you through implementation. It also reviews concepts such as volunteer and employee management, budgeting, public relations, emergency management, church growth and much more!