Installing and upgrading help

I recommend getting your Moodle 3.1.1 looking and acting the way that you want it to.  I was a big user of Formal White, and yes, it is not a standard theme now in Moodle.  You might be able to load it as a plugin.  But I got away from Formal White and now use the MORE base theme, with CSS modifications.  I gave a presentation about this at this year's MoodleMoot 2016 US.  Use any of my posted stuff as needed.

I cannot recall if there are any steps in going from 2.7 to 3.1.1.  Others here might be able to help you.

Make a backup of your moodledata, mysql database, and even your moodle before doing anything else.  Once you have a backup, you can start updating your Moodle 2.7.  You might actually consider doing this in steps, even if you don't have to.  I would upgrade to 2.8, then 2.9, 3.0, and finally 3.1.1.  Then you can either apply your 3.1.1 changes to this upgraded version, or you can move the old (but now 3.1.1) database and moodledata into your new Moodle 3.1.1.  I don't know if either is better, but maybe someone else here will have some ideas.

If you have never backed-up your Moodle, learn how to do this via SSH.  This will help you also move your Moodle if you end up doing so.  It's really not that hard.

Along the way, you will probably notice that "opcache" is recommended, and there are also some fixes to the mysql database.  These will become obvious to you.

I assume that you have SSH access to your server, right?  Are you fairly good at using it?

Since you have a Moodle 3.1.1 running right now, you at least know that this version works on your server.  Go to Site Admin | Server |Environment to see if you get all OKs.

I think that you will be pleased with Moodle 3.1.1.  Many good improvements.  I have Moodle 3.1.1+ (Build: 20160811) running right now (also about 20 courses.)

In my experience the migration from 2.7 to 3.1 has been pretty simple but it will likely depend how much you've customized Moodle with plugins and your selected theme.

Obviously check that you server meets the requirements. Typically this is also a good time to consider upgrading your server with the latest versions of MySQL/MariaDB, PHP/HHVM, Nginx/Apache (depending on your flavor).

In my opinion the safest and likely easiest way to upgrade is to clone your existing installation. Once you have a clone in place you never really have to worry about breaking anything and can fall-back to old installation. I typically clone to an external server for safety reasons to avoid deleting the wrong database or files by mistake. This also allows me to modify my windows host files and add the new server IP and then work on the clone using the same domain as I would on the production website. This way all your actions will be carried out in a temporary environment and at any time if you fail you can easily restart the process or simply revert back to your old installation (which would be untouched).

On your clone move your old installation to something like "moodle.old".

Then proceed to copy your config, themes, and plugins from your old installation to your new.

Instructions are pretty detailed here: https://docs.moodle.org/31/en/Upgrading

The mixed Moodle issues you are facing are likely from trying to copy a new moodle installation over your old moodle.

1. Move your old Moodle software program files to another location. Do NOT copy new files over the old files.

Following the upgrade guide is pretty critical to avoiding issues.

Another tip for anyone else reading this is the CLI provided by Moodle. There are a bunch of very useful command line tools provided by Moodle.

https://docs.moodle.org/31/en/Administration_via_command_line

I would not really recommend trying to "move" anything. You would be far better off completing an actual "upgrade".

I would consider testing first with the default theme provided. Once you are certain the upgrade is successful then consider moving to your previous theme.

Sorry I forgot to mention to make sure your plugins are also compatible with Moodle 3.1+