Hello there. I noticed your question and felt worried for you… expecially since nobody else has responded.
I hope you still have access to your old hosting!
You need to do a dump of your old MySQL database from your old server and then import it into the new server. The database will not be included in the files they copied accross. I presume they just copied the /var/www folder…?
A database more than just files in a folder, its some sort of Microsoft black magic and cant be copied the same way as files.
This link will get you started on how to do this
You may need to “Google” along the way.
Regarding moving your settings etc to the new server this is possible but could be problematic. Assuming you only have a few sites I would do a fresh install of ee and install each of them again manually. I always feel better doing that as its like a chance to start fresh.
Regarding the SSL you can copy this from one server to another but its not really how you are supposed to do it. It does work but the whole point of an SSL is to verify the identity of the server and there is every chance that this will come back and bite you down the line… Again its advisable to do this again manually.
The best advice I can give is copy and paste all the commands you do into a text editor with notes and keep a journal of everything you did so the next time its just a case of following your notes and pasting the commands into the new server. I use Evernote and keep very detailed records of every step i do on my servers. It makes it easier to rollback from errors and makes it MUCH easier the next time you build a server.
All that being said the host (assuming they know what they are doing) should be able to help moving the databases… You could also do a backup of the sites (which would include the databases). I think you have plenty of plugins for that. Backup Buddy comes to mind. Or Infinite WP if you manage several sites is worth the effort.
Anyway. Good luck. May the force be with you!