[SalesForce] How to use ActionSupport, ActionREgion, ActionFunction all together

I want to avoid using ActionRegion because it forces the user to enter data in required fields in other parts of the page form.

One of the main uses of the apex:actionRegion tag is to do the opposite of this, by sending only the elements that are children of the actionRegion with the request to the server. The visualforce parser will only enforce required validations on fields that are included in the request, and thus you can strategically place an actionRegion to get around unwanted validation.

In this case of the code in your question, the apex:actionRegion tag is just wrapping the wrong elements. Change it so the actionRegion only wraps the inputField.

<apex:actionRegion>
     <apex:inputField value="{!contact.AccountId}">
        <apex:actionSupport event="onchange" action="{!AccountPopulated}" rerender="accinfo, msgs"/> 
     </apex:inputField>
</apex:actionRegion>

Finally I came up with the following solution.

I use actionFunction instead of actionSupport and call it from JavaScript onChange handler.

<apex:page controller="requiredCont">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.6.2/jquery.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jqueryui/1.8.16/jquery-ui.min.js"></script>
<script type="text/javascript">
    function setTitleToValue(el) {
        el.title = el.options[el.selectedIndex].text;
        checkReq();
    }
</script> 

<apex:form >
    <apex:pageBlock >
        <apex:pageBlockButtons >
            <apex:commandButton id="update"  value="Save"/>
        </apex:pageBlockButtons>
        <apex:pageBlockSection >
            <apex:actionRegion >
                <apex:actionFunction name="checkReq" action="{!checkRequired}" reRender="otherFieldRequired"/>         
                <apex:selectList size="1" value="{!someAccount.Name}" title="{!someAccount.Name} - {!someAccount.Name}X" onchange="javascript:setTitleToValue(this)">
                    <apex:selectOptions value="{!names}"/>
                </apex:selectList>
            </apex:actionRegion>
            <apex:outputPanel id="otherFieldRequired">
                <apex:inputField value="{!someAccount.Site}" id="dependentField" required="{!isRequired}"/>
            </apex:outputPanel>
        </apex:pageBlockSection>
    </apex:pageBlock>
</apex:form>

public class requiredCont {

public List<SelectOption> names{
    get{
        if (names == null) {
            List<SelectOption> optionList = new List<SelectOption>();
            optionList.add(new SelectOption('a', 'a - aX'));
            optionList.add(new SelectOption('A', 'A - AX'));
            optionList.add(new SelectOption('B', 'B - BX'));
            optionList.add(new SelectOption('C', 'C - CX'));
            names = optionList;
        }
        return names;
    }
    private set;

}

public static Set<String> requiredPicklistValues = new Set<String> { 'A', 'B', 'C' };
public Account someAccount { get; set; }

public boolean isRequired  {get;set;}{isRequired=false;}

public PageReference checkRequired(){
    isRequired = requiredPicklistValues.contains(someAccount.Name);
    System.debug(LoggingLevel.ERROR, '@@@@ isRequired:' + isRequired );
    return null;
}
public requiredCont() {
    someAccount = new Account(Name = 'a');
}
}