I have published my website over the last year or so
several times with no problem, What does happen now though is that when you search it in google or what ever it shows the title of the 10th page of the website which is "media"and when you click on it it goes to that page not the home page. www.alabamaexpress.co.uk is the website but it shows as Alabama Express: media.
I thought that publishing it from the home page might fix it but it has not.
what page in RW is maked as your homepage? (will havw a house icon). Right click the page in RW and choose set as home page for the page you want. you Also have an untitled page next to your home page that should be deleted if your not going to use it.
Having a drop down is no issue off the home page. Just checked it out and it looks OK. Search engines such as google will need time to rescan your site in order for page indexing to be updated, so the changes you make right now may take days to get updated on the search engine.
The only way you can fix that other than waiting( that I am aware of) would be a redirect in an htaccess file on your web server.
Right now I tried www.alabamaexpress.co.uk/media and I get a 404 not found.
You could also submit your site to google, that may speed up them reindexing your site. If you have not done this before I would suggest you google “submit my site to google” there are some videos and help you can find, and here is a link to googles support on that subject: https://support.google.com/webmasters/answer/6259634?hl=en