[SalesForce] Bind checkbox value with controller boolean variable

Yes, you can use apex:param(s) inside the actionFunction to send values into the controller. The value attribute can be a controller variable as shown in the docs example or set with a javascript call.

Here is something I put in use recently-

<apex:actionFunction name="setTableData" action="{!setTableData}"
        immediate="true" rerender="pgMsgs, matrixTopPanel"
        oncomplete="oTable.fnAddData({!tableData});">
    <apex:param name="sd" assignTo="{!selectedDivision}" value="" />
</apex:actionFunction>

which is called from javascript with setTableData(j$(this).val());

Accessing controller properties uses the usual get & set syntax.

Set in a constructor and retrieved using the shorthand notation

public class YourController {
    public string varA { get; set; } // use the proper type

    public YourController() {
        varA = 'Some text';
    }   
}

or

Retrieved from the getNNN mechanism

public class YourController {
    public YourController() { }

    public string getvarA() {
      return 'Some text';  
    } 
}

or

Retrieved from a shorthand getter which calls a method in the controller

public class YourController {
    public YourController() { }

    public string varA { get { return doIt(); } }

    private string doIt() {
        return 'Some text';
    }
}

VF Page - JavaScript function - controller property reference will work with any of the above examples:

<script>
    function hello(){
        var theControllerValue = '{!varA}';
        alert(theControllerValue);
    }
</script>

The rendered source for the page, which you can go look at in the browser, will look like this after the substitution for the controller variable has been made:

function hello(){
    var theControllerValue = 'Some text';
    alert(theControllerValue);
}