As for the key signature, I am still seeing nothing about key in the pdf when I do this to remove all complexity:
\version "2.22.2"
\key d \major
{
a
}
put the first brace before \key and it will give you what you want.
Please work your way carefully through the Learning Manual. It is annoying to do so, but it will provide you the basis you need to succeed in LilyPond.
A score is a single musical _expression_.
You have two musical expressions in this file:
A) \key d\major
B) { a }
Each is wrapped in its own score contect by default. Score A creates no output. Score B creates the output of a single note, with no explicit key signature, so c major is assumed.
If you do
{
\key d \major
a
}
then you will have a single music _expression_, and you will get a single score that includes both the key signature and the note.
HTH,
Carl