A simple Google Apps Script to submit a Google Sheets powered timesheet: https://docs.google.com/spreadsheets/d/1D-YWjt_Zl8WrAuicmUKZsMzSpEOwv1vYL06dBdObyb4/

Code.gs

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// Example Google Sheets Timesheet: https://docs.google.com/spreadsheets/d/1D-YWjt_Zl8WrAuicmUKZsMzSpEOwv1vYL06dBdObyb4/

function submitTimesheet(e){
    var RECIPIENT_EMAIL_ADDRESS = 'SOMEBODYS_EMAIL_GOES_HERE'; // CHANGE ME - Where should this timesheet get sent?

    var error  = false,
        SS     = SpreadsheetApp.getActiveSheet(),
        ui     = SpreadsheetApp.getUi(),
        date   = SS.getRange("B5"),
        name   = SS.getRange("C2").getValue().split(' '),
        hours  = SS.getRange("Q5:R11").getValues(),
        start  = date.getValue(),
        end    = SS.getRange("B11").getValue(),
        _start = Utilities.formatDate(new Date(start), 'GMT', 'MMM. dd'),
        _end   = Utilities.formatDate(new Date(end), 'GMT', 'dd, YYYY');
    
    // Make sure email has been changed!
    if( RECIPIENT_EMAIL_ADDRESS == 'SOMEBODYS_EMAIL_GOES_HERE' ){
        ui.alert( 'Please change the Email Address (line 2 of the submitTimesheet() function)' );
        return;
    }

    // Rough validation of the email they changed, lol
    if( !validateEmail(RECIPIENT_EMAIL_ADDRESS) ){
        ui.alert( 'Invalid Email Address' );
        return;
    }

    // Make sure all hours are actually set
    if( !isComplete(hours) ){
        ui.alert( 'Your timesheet appears to be unfinished. Please make sure all start times have a corresponding end time!' );
        return;
    }

    var response = ui.alert('Submit Timesheet', 'Hi '+ name[0] +',\nAre you sure you want to submit the timesheet for ' + _start +'-'+ _end +'?', ui.ButtonSet.YES_NO);
    if( response != ui.Button.YES ){
        ui.alert( 'Submission has been halted. Please finish your timesheet and click the Submit Timesheet button again when you\'re ready.' );
        return;
    }
        
    // Send PDF via Email
    var result = sendSpreadsheetToPdf(0, SS.getName(), RECIPIENT_EMAIL_ADDRESS, name[0] + '\'s Timesheet: '+ _start +'-'+ _end, "Please find the attached timesheet:");
    
    if( result == true ){
        // Clear Content
        SS.getRange("E5:P11").clearContent();  // Wipe Current Times
        SS.getRange("S5:S11").clearContent();  // Clear Notes
        SS.getRange("U5:U11").clearContent();  // Clear Holiday
        
        // Update the Starting Date
        var d = new Date(start);
        date.setValue( new Date(d.setDate(d.getDate()+7)) );

        /**
         * Modify Forecast calendar
         * 
         * Move next week to current, postproximate to next and
         * clear postproximate
         */

        // IN OFFICE
        SS.getRange("D5").setValue( SS.getRange("E15").getValue() ); // Mon
        SS.getRange("D6").setValue( SS.getRange("G15").getValue() ); // Tue
        SS.getRange("D7").setValue( SS.getRange("I15").getValue() ); // Wed
        SS.getRange("D8").setValue( SS.getRange("K15").getValue() ); // Thu
        SS.getRange("D9").setValue( SS.getRange("M15").getValue() ); // Fri
        SS.getRange("D10").setValue( SS.getRange("O15").getValue() ); // Sat
        SS.getRange("D11").setValue( SS.getRange("Q15").getValue() ); // Sun

        // PTO
        SS.getRange("T5").setValue( SS.getRange("F15").getValue() ); // Mon
        SS.getRange("T6").setValue( SS.getRange("H15").getValue() ); // Tue
        SS.getRange("T7").setValue( SS.getRange("J15").getValue() ); // Wed
        SS.getRange("T8").setValue( SS.getRange("L15").getValue() ); // Thu
        SS.getRange("T9").setValue( SS.getRange("N15").getValue() ); // Fri

        // POSTPROXIMATE WEEK TO NEXT WEEK
        SS.getRange("E15:Q15").setValues( SS.getRange("E16:Q16").getValues() );

        // CLEAR POSTPROXIMATE WEEK
        SS.getRange("E16:Q16").clearContent();
        
        ui.alert( 'Thanks '+ name[0] +'! You have successfully submitted your timesheet for '+ _start +'-'+ _end +'!' );
    } else {
        ui.alert( 'An error occurred when trying to send your timesheet.' );
    }
}

function sendSpreadsheetToPdf(sheetNumber, pdfName, email, subject, htmlbody){
    var SS       = SpreadsheetApp.getActiveSpreadsheet(),
        ui       = SpreadsheetApp.getUi(),
        sheetId     = SS.getSheetId(),
        url_base = SS.getUrl().replace(/edit$/,'');

    var url_ext = 'export?exportFormat=pdf&format=pdf'     //export as pdf
        + (sheetId ? ('&gid=' + sheetId) : ('&id=' + spreadsheetId)) 
        + '&size=A4'           // paper size
        + '&portrait=false'    // orientation, false for landscape
        + '&fitw=true'         // fit to width, false for actual size
        + '&gridlines=false'   // hide gridlines
        + '&fzr=false';        // do not repeat row headers (frozen rows) on each page
        + '&sheetnames=true';  // show sheet names
        + '&printtitle=false'; // hide print title
        + '&pagenumbers=true'; // Hide page numbers

    var options = {
        headers: {
            'Authorization': 'Bearer ' + ScriptApp.getOAuthToken(),
        }
    }

    var response = UrlFetchApp.fetch( url_base + url_ext, options );
    var blob     = response.getBlob().setName(pdfName + '.pdf');

    if( email && validateEmail(email) ){
        var mailOptions = {
            htmlBody: htmlbody,
            attachments: blob
        }

        MailApp.sendEmail(
            email, 
            subject, 
            "html content only", 
            mailOptions
        );

        return true;
    } else {
        return false;
    }
}

// "Total Hours" Formula: =if(isodd(countblank(E5,G5,I5,K5,M5,O5)),"…⏳…",sum(G5-E5,K5-I5,O5-M5,U5*K22))
function isComplete( checks ){
    var re = /\d*:\d*/;

    for( var i = 0; i < checks.length; i++ ){
        if( checks[i].indexOf('…⏳…') > -1 || !re.test(checks[i]) ){
            return false;
        }
    }

    return true;
}

// [email protected] - No *true* front end validation for this
function validateEmail( email ){
    var re = /\S+@\S+\.\S+/;
    return re.test(email);
}

function isEven( value ){
    if( isNaN(value) )
        return false;

    return ( value % 2 == 0 ) ? true : false;
}

function isOdd( value ){
    if( isNaN(value) )
        return false;

    return ( value % 2 != 0 ) ? true : false;
}

function onEdit(e){
    // Prevent "PTO and In-Office" from being selected together.
    var SS       = SpreadsheetApp.getActiveSheet(),
        forecast = SS.getRange('forecast'),
        range    = e.range,
        row         = range.getRow(),
        col         = range.getColumn(),
        value    = range.getValue();

    // Are we in the Forecast range?
    if( col < forecast.getColumn() || col > forecast.getLastColumn() || row < forecast.getRow() || row > forecast.getLastRow() )
        return;

    // What's the comparison? Odd cells compare to Next, Even to Previous
    var compareTo   = ( isEven(col) ) ? col - 1 : col + 1,
        compareCell = SS.getRange(row, compareTo, 1, 1);

    // If this value is TRUE, set it's comparison cell to FALSE,
    if( value == true )
        compareCell.setValue( false );
}