1. She is eating an ice cream while reading the newspaper.
  2. She is eating an ice cream during reading the newspaper.
    Are these two sentences correct?
    If they are, then are ‘while’ & ‘during’ used as preposition?

‘While’ is a conjunction and cam start a new clause (sentence) or verb phrase and ‘during’ is a preposition and so 1 is the correct sentence.

Look at this: During the afternoon she was eating an ice-cream while (she was) reading the newspaper.

Why “the newspaper”? Is it possible to say “a newspaper”. Thanks

Both are possible.

Please see this recent thread … tml#430005

an ice-cream or ice-cream??? … =ice-cream
It can be both countable and uncountable (see: n [C or U]):

An ice-cream
A cone/scoop of ice-cream.