The fields of characteristic $p$ are such that "$p=0$" by handwaving. Therefore, if $1=0$, the only field you can expect is the zero field, which is indeed, as you stated, a bit strange, for it is the only field with this property. For every other field, $1 \neq 0$.
(EDIT : You can interpret my word "expect" in "the only field you can expect" this way : the definition of field that allows $1=0$ only adds the zero field to the possible fields, even though it is non-standard to do so, so we usually assume $1 \neq 0$ to get rid of this case. See the discussion in the comments for more details.)
Usually people do not consider $1$ as a prime, for it does not generate a prime ideal in the ring $\mathbb Z$. Now this again is a matter of definition ; we define the prime ideals those who satisfy some property and are not the whole ring. There are many other reasons why $1$ is usually not a prime, and you just found one of them. $1$ behaves significantly differently than the non-$1$ primes, so it is natural to leave it aside.
Hope that helps,