What is the significance of transcendental vs algebraic numbers and functions?

Is one more meaningful than the other? Is one able to be proven to a stronger degree?

For incalculable transcendental numbers, can anything about them be proven, or is it all conjectured?

Is there any way to contract the uncountably infinite digits of an irrational mumber, or deal with them such in a way that an exact measurement can be achieved, rather than rounding off at some number of decimal digits?