What does lowering the key of a song by a semitone exactly mean? If a song is for example in F major, and you lower the key by a semitone, do you lower every note by a semitone? This becomes E G♭ A♭ A B D♭ E♭ E, which sounds the same as E major, but E major doesn't have flats but sharps. So in what key would it be? Or do you add one more flat in the key signature, so that F major becomes B♭ major?
Yes, that is what lowering the key of a song by a semi-tone means. You transpose every note down one half-step. Changing from the key of F major to the key of Bb major would actually be raising the key by a fourth (five semitones), although it's clearer to say "transposing up a fourth".
Gb is enharmonic to F#, Ab to G#, etc. If you replace each flat with its enharmonic sharp, you'll end up with the key signature for E major.
If you wanted, you could instead spell "E" as "Fb", also "A" as "Bbb" and "B" as "Cb". Then you would have a key signature for Fb major, which is enharmonic to E major. Of course, no one ever does this, but I thought it might help explain why your brain seemed to go in a valid direction and still was confusing. Instead of changing some letter names and then adding flats to other letters, you would either change none of the letters and add a flat to all of the existing letter names, or change all of the letter names down by one and add sharps when necessary to make sure the new key is major. That's a general guide to transposing keys.
You're on the right track, but adding to Todd's answers each note gets dropped down by a semitone, it puts the song into key E. Thus each new note must belong to that key, and that key contains one of each letter name.
Looking at your idea, you'll see there's no F, two As and two Es. It'd be a nightmare to try to re-write the music that way!. So, the notes will be from E F# G# A B C# D#. Note - one of each. What's actually happened is that each letter name before has dropped by one - F to E, Bb to A, D to C# etc. Yes, effectively each original has been flattened, but in some keys (E is one) the notes take on new names. Like that 'Db' gets known as 'C#'.
Certainly what you do not do, is add another flat. That would sort of put it into Bb, but if the notes didn't change it would only have one affected note, and if they did change down a semitone, you'd end up with key E, but with a flattened 5, which wouldn't work at all.
To work out what to replace each note by, it is better to lower each note by the same interval rather than some number of semitones. In this case, you want to transpose from F to E. This entails lowering by a minor second. (Because each note is lowered by a second, that means that it gets replaced by a note whose letter name is one letter further down.)
This means that notes get replaced as follows:
E --> D#
D --> C#
C --> B
Bb --> A
A --> G#
G --> F#
F --> E
The same applies to any chromatic notes, too. Thus a D flat would become C, whereas a C sharp would become B sharp.