Update Panel is one of the most simplest way of implementing AJAX on your site. On your local system, you may tend to drop half a dozen Update Panel and it will work smoothly.
However, the same Update Panel will go through the entire page life cycle and may create performance bottlenecks in live environment. The use of update panel will also increase the overall page size.
Page Methods are far better as instead of posting back and then receiving HTML markup to completely replace our UpdatePanel’s contents, we can use a method to request only the information that we’re interested in.
Answering to your questions:
Understanding is correct?
Your understanding of ASP.NET AJAX/JQuery is completely correct.
Issue in accessing controls in code behind files?
Only ASP.NET ajax will allow you to access the controls in code behind files. However that comes with a performance bottleneck. Once you get used to other Javascript frameworks such as JQuery, you will always find a workaround of resolving this issue.
My suggestion is to use ASP.NET AJAX on site where performance requirements are not too high. You can also use it on intranet application.
But when it comes to performance, use a light-weight javascript library.