The same holds true for that Seamonkey browser you were looking for. When you're trying to get software that is not in the repositories you literally have to add another repository (called a PPA for Personal Package Archive), which is why we need the "command line" (terminal).
Unlike Opera (no offense, @Vera ), Seamonkey is fully open-source, free software and it will be completely familiar to you, as a Firefox/Thunderbird user):
Open the terminal and type this:
sudo thunar
and enter your root password.
This opens the File Manager in "super user" (Administrator, or in Linux, 'Root') mode. You have to be root in order to edit the software sources file.
Navigate to etc/apt/sources and find the file named sources.list. Right-click on the file and open it with Text Editor.
Add this line to the the bottom of the sources.list file:deb http://downloads.sourceforge.net/project/ubuntuzilla/mozilla/apt (http://downloads.sourceforge.net/project/ubuntuzilla/mozilla/apt) all main
Save and close the file. Then run the following command to import Ubuntuzilla public key to your keyring so that
the integrity of packages downloaded from this repository can be verified by APT.
In the terminal type:
sudo apt-key adv --recv-keys --keyserver keyserver.ubuntu.com 2667CA5C
And thensudo apt-get update
(wait for it to update, and when it's finished,
sudo apt-get install seamonkey-mozilla-build
This will not only install Seamonkey for you, but will also add it's PPA to your software sources
so that it will be included in future updates.
Opera probably has a PPA too, but be careful about adding a whole bunch of PPAs to Linux Lite.
I really hope this helps.