NOTE1: One pre-requisite of this, is that you need to have some knowledge in how Shell Scripting Works. and that you do understand what parts of the script you can modify. i would assume so in this tutorial

NOTE2: You have to change part of the script that is in "green font"

STEP1: Download the Palemoon Tar Version directly from the Palemoon Website

STEP2: Extract the palemoon tar.gz file on your "/home/yourusername" folder

STEP3: Copy The Entire Script Below and Modify Portions of it that you like to modify. but as it is, the script below works in properly installing and getting the icons to properly display on your menu.

STEP4: Once you've reviewed the script below and customized it to your own liking, you can just make it executable. (mine, i named it "InstallPalemoon.sh") remember to place the script in the same folder as the palemoon tar gz file.

#!/bin/bash
echo "Copying Palemoon To /opt Directory Please wait..."
sudo cp -r /home/ronald/Downloads/palemoon /opt/palemoon  && echo "Palemoon successfully copied to /opt directory!"

echo "Copying Icons Please Wait..."
sudo cp /opt/palemoon/browser/chrome/icons/default/default16.png /usr/share/icons/hicolor/16x16/apps/palemoon.png
sudo cp /opt/palemoon/browser/chrome/icons/default/default32.png /usr/share/icons/hicolor/32x32/apps/palemoon.png
sudo cp /opt/palemoon/browser/chrome/icons/default/default48.png /usr/share/icons/hicolor/48x48/apps/palemoon.png
sudo cp /opt/palemoon/browser/icons/mozicon128.png /usr/share/icons/hicolor/128x128/apps/palemoon.png
echo "Icons Successfully Copied!"

echo "Creating File Configuration For The Menu Entry"
touch palemoon.desktop
echo "[Desktop Entry]
Version=1.0
Name=Pale Moon Web Browser
Comment=Browse the World Wide Web
Keywords=Internet;WWW;Browser;Web;Explorer
Exec=/opt/palemoon/palemoon-bin
Terminal=false
X-MultipleArgs=false
Type=Application
Icon=palemoon
Categories=Network;WebBrowser;Internet
MimeType=text/html;text/xml;application/xhtml+xml;application/xml;application/rss+xml;application/rdf+xml;image/gif;image/jpeg;image/png;x-scheme-handler/http;x-scheme-handler/https;x-scheme-handler/ftp;x-scheme-handler/chrome;video/webm;application/x-xpinstall;
StartupNotify=true" > palemoon.desktop
sudo cp palemoon.desktop /usr/share/applications/palemoon.desktop
echo "Menu Entry File Successfully Created!"
echo "Refreshing Menu Entry"
sudo gtk-update-icon-cache -f /usr/share/icons/hicolor
echo "Menu Entry Successfully Refreshed"
echo "Palemoon Successfully Installed!"
sudo rm -f ./palemoon.desktop

#SCRIPT ENDS HERE

STEP 5: Just make your script executable. again make sure your script is in the "/home/yourusername/Downloads" directory. where you have the extracted palemoon folder. just run your script and after that, you should be able to see Palemoon Browser appear on your menu. you can also create a shortcut of it on your panel