For 1., just prove directly that $x^2 + y^2 -1$ is a prime element of $\mathbb Q[x,y]$. If you want to, you can use Gauss's lemma, which reduces you to showing that it is irreducible, which is particularly simple. (But you can also prove it is prime directly pretty easily --- you will more or less recover the proof of Gauss's lemma in this special case.)
For 2., you need to rationally parameterize the conic $x^2 + y^2 - 1$. This is a standard part of the theory of pythagorean triples. (See the wikipedia entry; note that the quantity called $m/n$ there is your $t$.) The same geometric argument shows that any conic can be rationally parameterized if it contains at least one point defined over the ground field.