[SalesForce] How to create a Tab that shows sub-menu on Click in salesforce

Change the switchType on your apex:tabPanel to "client".

PS This How can you make a visualforce graph display correctly within a TabPanel? looks like it covers the same sort of page and the recommendation is to stick with switchType of "server". Perhaps you can add to your question what is problematic about the refresh you are getting at the moment.

Well, you've kinda answered your own question with this:

Each of the tab contains an actionFunction with the same name "methodOneInJavaScript"

Let's say, if you create a simple page with several js functions with the same name:

function testFunction() {
    alert('1st');
}

function testFunction() {
    alert('2nd');
}

function testFunction() {
    alert('3rd');
}

...the last one will override the previous ones. Meaning, whenever you call testFunction(), the implementation defined in the last one will be invoked (in this example, the page will alert the message '3rd').

In your case, what happens is this - you click the Refresh in the first tab, it invokes function methodOneInJavascript(), but the last one defined in the entire page, which is the one in the "AR Notes" tab:

<apex:actionFunction action="{!refreshTab}" name="methodOneInJavascript" rerender="ART" oncomplete="getDataTable('ART')">
    <apex:param name="callingTab" assignTo="{!TabName}" value="ART" />    
</apex:actionFunction>

This in turn invokes getDataTable('ART') upon completion.

The best way to convince yourself is to inspect the source of the page (right-click > View Source) and search for all occurrences of "methodOneInJavascript". I'd advise to rename the methods accordingly.