Is √a²+√b² positive or negative

It's positive or 0.

One simplification is | a | + | b |

√n is defined as the positive square root of n.

The function "√" is supposed to be positive square root unless mentioned otherwise. Both signs are not taken because then it will be multivalued hence not a function.
According to this convention, √a²+√b² is positive or zero, when both a and b are zero.

Square roots can be positive or negative. However, convention is that a radical without a sign is to be considered to be the positive root. Therefore, √a²+√b² will be non-negative (it is positive or zero).

If you are talking about the principal square roots, the sum of two positive numbers is always positive. If you're not talking about the principal roots, the question is indeterminant.

√a^2+√b^2
a+b

Could be positive or negative

If a=5 and b=10, then a+b=5+10=15.
If a=-5 and b=-10, then a+b=-5-10=-15.

You can take the radical of a negative number and square it because they cancel out. There's nothing wrong with saying √(-1)^2.

Remember that when you take an even root, it's always plus or minus that number.