Consider the following suggestions:

When looking at the function stripRows, note that the variable rows is referenced without being defined. It seems like you intended to include something like

var rows = document.getElementsByTagName('tr');

at the beginning of the function.

Additionally, within the same function, there is a line that reads

} else if (rows[i]/2 != 0 && !rows[0]) {

. It appears that instead of this, you might have meant } else if (i%2 == 0...){. In fact, it may be more effective to rewrite the entire conditional statement as follows:

if( i%2 == 0 ) {
    addClass(rows[i],'even');
} else {
    addClass(rows[i],'odd');
}

These adjustments should aid in addressing the errors present and set you on the right path towards resolving other similar issues in your code.

You do not require the use of getElementsByTagName. The table and table section elements have a property called rows, which represents their child rows. Therefore, you can implement something like this:

function stripeRows(tableSection, oddClass, evenClass) {
  var rows = tableSection.rows;

  for (var i=0, iLen=rows.length; i<iLen; i++) {
    rows[i].className = i%2? oddClass : evenClass;
  }
}

Please note that "tableSection" can refer to a table, tBody, tHead, or tFoot element. For simplicity in this example, I have used an "addClass" function to add the class.

Edit

A few comments on the code:

function stripemeTable() {
    // obtain the <tr> tags:
    var stripemeRows = document.getElementsByTagName('tr');

This fetches all tr elements within the document, but it might be more efficient to do this table by table.

// IF the tableID is NOT stripeme, exit the stripemeTable function:
if (!document.getElementById || !document.getElementsByTagName || !document.getElementById('stripeme'))  {
    return false;

In this snippet, there is a check for getElementsByTagName, although this method has already been utilized in the preceding line. If testing for getElementById is necessary (which is uncommon now that most browsers support it), consider:

     var stripeMe = document && document.getElementById && document.getElementById('stripeme');
     if (stripeMe) {

The assumption here is that the rows belong to the table 'stripeMe' without confirming it. If striping the rows only when they are within a table element with an ID of 'stripeMe', then proceed accordingly. It would be preferable to use a class instead. The else statement becomes unnecessary, simply continue as follows:

// ELSE loop through the rows and apply the stripeRows function:  
} else {
    for (var i = 0; i < stripemeRows.length; i++) {
        stripeRows('stripeme','pri-stripe','sec-stripe','over-stripe');
    } // end of the rows FOR loop 
} // end of the tableID test IF/ELSE

In place of this approach:

     var stripemeRows = stripeMe.rows;
     for (var .... ) {
       ...
     }

If you specifically want to stripe the rows inside the body, separate the header rows into a tHead element and stripe only the rows within the tbody, for example:

 var stripemeRows = stripeMe.tBodies[0].rows;

Hello @Jason-McCoy, after reviewing the requirements thoroughly, I am confident that this solution fulfills all the criteria you have set. By starting the initialization of i at 1 instead of zero, the first line is preserved. Additionally, the stripeRows function does not perform any direct selections based on the elements passed to it. The striping functionality should work properly with this code, although some bugs still remain in the onmouseover section. Please pay close attention to that during the implementation.

addLoadEvent(stripeme);

function stripeme() {
    if (!document.getElementById || !document.getElementsByTagName) { 
        return false;
    }

    // If the tableID is NOT "webcourses", exit the webcoursesTable function:
    if (!document.getElementById('webcourses') && !document.getElementById('stripeme')) {
        return false;
    }

    // Retrieve the <tr> tags:
    var rows = document.getElementsByTagName('tr');

    for (var i=1; i < rows.length; i++) {
        stripeRows(rows[i],(i%2==0)?'pri-stripe':'sec-stripe','over-stripe');
    } // end of the rows FOR loop 
} // end of the stripemeTable function

function stripeRows(row,rowClass,over) {
    addClass(row,rowClass);

    // Get the current className of the table rows:
    var oldClassName = row.className;

    // Handle the onmouseover event for the table rows:
    mouseOver = row.onmouseover;
    mouseOver = function() {
        addClass(row,over);
    } // End of the onmouseover anonymous function

    // Handle the onmouseout event for the table rows:
    mouseOut = row.onmouseout;
    mouseOut = function() {
        addClass(row,oldClassName);
    } // End of the onmouseout anonymous function
} // End of the stripeRows function 

function addClass(element,theClass) {
    // If the table row does not have a className, append the required className:
    if (!element.className) {
        element.className = theClass;
    // Else append the required className to the className(s) that are already assigned to that row:
    } else {
        var newClassName = element.className;
        newClassName += " ";
        newClassName += theClass;
        element.className = newClassName;
    } // End of the className IF/ELSE  
} // End of the addClass function

function addLoadEvent(func) {
    // Assign the window.onload function to a variable:
    var oldonload = window.onload;

    // If the window.onload does not call a function:
    if (typeof window.onload != 'function') {
        window.onload = func;
    // Else set the window.load:
    } else {
        window.onload = function() {
            oldonload();
            func();
        } // End of the oldonload anonymous function
    } // End of the typeof IF
} // End of the addLoadEvent function

For your CSS styling:
You can either include this in both striped-tables.css and abstraction.css, or link to the same CSS file in both HTML files.

.pri-stripe {
    background-color: green;
}

.sec-stripe {
    background-color: blue;
}