For question 2, you can look for surfaces of Picard number 1. One way to find such surfaces is to reduce modulo p and show that the reduction has Picard number 1 by computing its L-function. This will only work for odd degree. For even degree, you may need to compute the reduction modulo two primes with Picard number 2 but incompatible lattices so the Picard number of the original surface is 1 (this is a trick due to R. van Luijk). These computations are not easy and there is some work, e.g. by Kedlaya, to make them feasible.
Question 3 is false. You can easily write down smooth surfaces of any degree containing a straight line.
Off the top of my head, I don't have an example for your question 1, which is unconditional. But, if believe the ABC conjecture in its multiple summands variant, then diagonal hypersurfaces give examples (e.g. $x^n+2y^n+3z^n=6w^n, n$ large).