Is it possible to express every decimal as a fraction (without rounding)? Why or why not?

Yes, you would just have to change the denominator. For example,2 would be 2/10 and. 201 would be 201/1000. A good rule of thumb is to use place value.

. ____2____ _____0_____ ______1_____
      tenths     hundredths     thousandths

Since the last place value is thousandths my denominator is going to be in the thousands.  

If the decimal ends, then always yes.
When you read the decimal, you call it some number of
tenths, or hundredths, or thousandths, or millionths etc.
and what you call it when you read it consists of the same
words you use when it’s written as a fraction.

If the decimal never ends but just keeps going on forever,
-  if the same group of digits repeats over and over, (as they do in the
decimal form of 1/3, 1/7, and 1/9), then the decimal can be expressed
as a fraction, even though the decimal never ends.
-  if a group of digits doesn’t repeat, (as in ’pi’ or ’e’ for example), then
the decimal is an irrational number  The definition of an irrational number
is one that can’t be expressed as a fraction.