Unfortunately, there is no "direct" way of using your MySQL database in ArcGIS, as MySQL is a non-supported database platform in the ESRI product line. This means you can not create ESRI enterprise geodatabases in a MySQL database.
You might consider converting to a PostgreSQL database, if you want to take full advantage of all options the ESRI enterprise geodatabases have on offer, and stay with Open Source.
Other than that, I guess you will be stuck with exporting the data to some file format the ArcGIS Viewer for Flex supports, possibly shapefile or maybe through some more steps using ArcMap to a file geodatabase... Be aware that shapefiles may truncate field names though due to limitations in the file format.
I have to little experience with MySQL and ArcGIS Viewer for Flex to guide you further on this. I think you will need to wait for some responses by other forum members.
2) Trying to Publish the REST Endpoint that exposes the MySql Spatial data as xml or json(xml preferable)
Is there any other best idea to do this please revert me if any of you guys are working with similar kind of work process
Can't comment on the specifics of web services and REST, as I have no experience with it, but if you are able to expose feature data as JSON, you might be able to make some good use (possible or most likely after your own code customization) of ESRI's new Spatial Framework for Hadoop and the Geoprocessing Tools for Hadoop, as it contains functions for converting JSON data to and from geodatabase Feature Classes. That is probably not a good end-point for you, but since the code is Open Source, you may be able to convert to some other format more suitable to your requirements on the client side.
I guess this is a rather involved procedure though, some people with more experience in this area can probably come up with easier solutions and references to useful code.
For some insights into what the Spatial Framework for Hadoop is, see my PDF and all the references therein here.