As of Flash to read/write to a database (MySQL or MSSQL)

would recommend to have a look in the direction of the AMF3.
It is native to flash/flex the transmission format is given.
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On the server side if the php Zend_Amf (http://framework.zend.com/download/amf) or php-amf3 (http://sourceforge.net/projects/php-amf3/), the latter is much faster then if pyAMF python in conjunction with django, twisted or anything else to taste )) about java not aware of.

Coldfusion promoted Adobo server platform for web applications based on J2EE.
In fact to communicate with the database in any way will have to write a server component handles the client requests the swf and produces various operations with the database. Essentially the choice comes down to the programming language which is most convenient for you.

"As an swf object, which executes on the client side will be able to read/write to the database that resides on the server side?" — here is the right question.

For transport to use HTTP or XML and they need the data to wrap and unpack on the server side.
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Quite versatile approach in any case, given that in Flash there are direct functions work on HTTP/XML

Not a big expert on Flash, but doesn't he have quite highly developed means to access network services? If all you need to do is provide access for the application to the DB, just raise the DBMS external interfaces of the server (if it is about the client application with a Central database) or local (if it is about local application) and please :) it is even Possible for Flash (Actionscript?) there are a variety of adapters, simplifying and abstracting database and, for example, writing SQL queries is not necessary.
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But just think, if you really need access to your database or you need data that is stored, and/or the ability to record them, but with additional logic, for example, advanced authentication, and authorization? Then it is necessary to introduce a three-tier architecture — client application -> app server -> database

There is already quite a lot and rightly said above. Maybe my experience will be useful. Now in one project I use direct SQL queries from a video through php script (in the elementary variant — 10 lines of code). A PHP script handles the incoming request and gives the information back to the movie in JSON. I can say that it is very convenient and easy way of sql queries is enough to perform complex samples, extra data between the client and the server is not running. In flash, we get an array or an object with the selection results. In my sql2flash class, the queries look like this: sql2flash("select * from table", slqHandler), where sqlHandler handler query results. If interested, I can send the code.

What do you mean "Them out can be elementary to get"? Especially if they are generated programmatically. Write about it in Adobe.
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How does an online game? I wrote above that it is necessary to understand the feasibility of this solution. I like thus pull data from ~10 interconnected databases, which are exclusively public information. I want to see this monstrous bike with an API that you offer, especially if the data structure can vary.
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And if you need to access sensitive data — no one bothers to write a separate php script.
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And again this database with passwords it is closed,-d-TA. Well, if people do not understand, he is not one tutorial will not help to protect the system aobenni in the case of such a fairly trivial decision as flash<- >mysql.