What are some common ways to open a real estate office?

The obvious solutions are to rent the office and pay sales people, I am more interested in other means such as offering shares in the company or some form of salesperson buy-in.

Assuming all broker relationship and relevant real estate law are followed, which methods are most common and benefit a single party as the head, president, manage, whatever title is most relevant.

Also, what is the common practice to deal with the allocation of clients that do not come in with an agent preference?

You seem to be a little short of real estate brokerage knowledge. First of all, In most states you will require a Broker's license, or have someone working as your Broker. That being said, you do not pay Realtors, as such. You earn commissions by having your office's listings sold or by selling other brokerages' listings. The commissions earned are then shared with your listing or selling agents according to the agreements you and they have signed together upon their employment. The splits may very from 50/50 to 95/5, or combinations thereof.
Offering shares or buy-ins is much more complex, and I would suggest keeping it simple to start. I would also recommend that you enquire about some of the franchises available, although they will likely require a fair amount of capital and experience on your side before getting into serious discussions.