How to build a web site on which clients can send and receive bitcoins without an account?
A payment gateway may have restrictions on the type of business category, and online gambling may be a category they don't support.

The process is pretty simply.

Supply a new address for each deposit.  When a payment arrives to that deposit address the customer account is credited.  

A payment is considered final, generally, once it has six confirmations.  That may be a long time for a customer (an hour, or longer even) -- some sites use one, or three confirmations but then have withdrawal restrictions so that newly deposited funds are not withdrawn until the original deposited funds have six confirmations.

But this business deals with money. Any business that deals with money, whether dollars, euro, gold, or bitcoin, should not be operated without giving appropriate attention to the factors that require attention.  With Bitcoin being a non-reversable payments system, IT Security becomes among the most important areas of the business.

Learning how to construct a payment transaction (sendtoaddress using bitconid) or receiving a payment (poll bitcoind using listtransactions, or -walletnotify) should be your last concern at this point.  Figuring out how to manage a wallet, using a cold wallet and hot wallet, for example, and making sure you have backups, etc., are a higher priority.

This experience might be something you hire out for rather than solicit from a Q&amp;A site.