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Für Elise intro starts with E D♯ E D♯, Wikipedia says the key is Am.
What scale does it fit? Is it like a blues scale where D♯ is a flat 5?
Why D♯ is used in the sheet music? Is it easier to read compared to E E♭ E E♭ or is there a certain naming convention?
It is common to use notes that are not in the scale to add color. It's called chromaticism, from the ancient Greek word for color.
Think how composers use a G# instead of a G in A minor, for example as a part of an E chord. A semitone creates more tension and the tendency of G# to resolve to (go to) A is more powerful. This is called a chromatic approach note. But since this particular usage is very common, we usually label it the "harmonic minor scale".
Similarly, using an F# to avoid the melodically awkward (to western classical ears) augmented second between F and G# is also quite common. Then we label it as the "melodic minor" scale.
These two so-called alterations are very common to warrant a new name. But there are many other possibilities.
The D# here is a quite common example. It's a chromatic approach note to E. A few notes later Beethoven uses a D natural, notice how that strong tendency of resolving to E is now mostly gone.
The reason it's labeled D#, instead of Eb is this. It's not a flattened fifth, it's a sharpened fourth. The fifth (E) is still there, this one is a degree lower, so it's a (sharpened) fourth. A flattened fifth (Eb) would replace the E and would tend to resolve to D, because just like the sharpened notes tend to resolve upwards, flattened notes tend to resolve downwards.
Granted, it mostly concerns classical music. Blues is a different beast. The flattened fifth and sharpened fourth can be used interchangeably. The convention is still to notate them with a sharp for upward moving lines and with a flat for downward moving lines but it's not used consistently.
That blue note, after all, does not even exist on a piano when properly played or sung. It's generally flatter than D#/Eb when sung or played on a fretless instrument or on an instrument that can do bends. It's an alien coming from a different musical culture, a square peg that we try to squeeze in to the round hole we have: The so-called equal temperament.
Yes, Ludwig started the Blues. Only kidding, but that note may be considered as part of a secondary dominant. The dominant of A minor is E, maj. or min. The dominant of that is B, with a D#. That's one way to look at it. Another is to say one is not just restricted to writing the notes that are only found in the original key. That's actually quite restrictive.
As Lee states, writing out, having to naturalise every other E is a pain - reading it worse, so D# wins.
I wonder if anyone else out there considers the first 8 notes to be a long anacrusis? The tune proper starting on the A ?
The D# could have been a D as well, but a half-step difference creates stronger tension, which is exactly what the composer was (presumably) going for. The same thing often appears in chord schemes, as explained in Tim's answer to a question that I asked a while ago.
As to your second question: indeed, E and D# are easier to tell apart (and easier to note down) than E and Eb. Imagine what that measure would look like with all those repeated accidentals.
It is just a small chromatic motif in the melody. It starts with the dominant chord (E Maj) chord and then moves back to the a minor chord where there is several variations in the picking from the a minor chord. It is used somewhat similar to an upbeat.
The D#/Eb note is used in much the same way a flat fifth/sharp fourth is used in blues or jazz. Simple passing tone that builds tension. Somewhat dissonant so as to make the resolve to the so called "right" notes sound better. Blues and jazz musicians regularly disregard the technical concept of the right notes and use the wrong ones to make the right sound even more right. Sometimes within small nuanced movements, in a way key becomes irrelevant. Music is sound. If it sounds good, it is.
I think of it like this. A bit unusual because these are the very first notes of the piece, but still, this is how I interpret it. The piece is clearly Am. But, this part of the song (which is repeated) is a momentary modulation to the dominant key E major. The d# is the third of a B or B7 chord which is implied, though not actually present. (dominant of the dominant). V/V -> V
Imagine playing the melody with these chords, using the same rhythm as the melody: E B E B E B dm C am (use a voicing of the B chord that emphasizes the d#, the third B chord, use the b melody note)
