Is square root of every non square number always irrational? find the smallest natural number which divides 2205 to make its square root a rational number.

Yes, all square roots of non square numbers are irrational. That means that the sqrt of a whole number will never be a fraction, but always an irrational number or an integer.

Second question, not entirely sure who "it" is...

The divisors of 2205 are: 1, 3, 5, 7, 9, 15, 21, 35, 45, 49, 63, 105, 147, 245, 315, 441, 735, 2205. 

Trying them out small to large:

√(2205/3) =  27,11088... irrational 
√(2205/5) = 21 rational!

So 5 is a divisor because 5*441 = 2205. Also √441=21 and 21 is rational.