Do accidental notes have their own name?

The answer is yes, such a system exists. It is called "Integer Notation", and instead of a naming scheme, it is actually a numbering scheme, from 0 to 11. In other words, C=0, C#/Db=1, D=2, etc..., up to B/Cb=11. To avoid confusion, 10 and 11 are often written as 't' and 'e' respectively.

1. What are accidental notes called?
2. What is the name of an accidental?
3. How is an accidental written?
4. Will a note considered accidental fit into the key signature?

What are accidental notes called?

In music, an accidental is a note of a pitch (or pitch class) that is not a member of the scale or mode indicated by the most recently applied key signature. In musical notation, the sharp (♯), flat (♭), and natural (♮) symbols, among others, mark such notes—and those symbols are also called accidentals.

What is the name of an accidental?

The most commonly used accidentals in music are the sharp (♯), the flat (♭), and the natural (♮). These accidentals raise or lower a pitch by a half-step, making the pitch either higher or lower than it was before the accidental.

How is an accidental written?

The accidental changes the pitch, so that the note is either higher or lower than the original natural note. Accidentals are written in front of the notes, but in text, accidentals are written after the note names.

Will a note considered accidental fit into the key signature?

Accidentals can be sharp, flat, or natural notes depending on the context of the key. For example, if we're in the key signature of G major which features the notes G, A, B, C, D, E and F#, any use of an F natural note would be considered to be an accidental.