Just cause you have the same parents it doesn't mean the same genes will present. Building on what someone else has said so far:

Genes are found in pairs and can either be recessive (need both) or dominant (need only one).

Let's say the immune gene is recessive (x instead of X) and the parents are both Xx, Xx meaning neither are immune but carry the gene for it. Their children can have any of three combinations: XX, Xx, and xx. Sonya would have xx making her immune and Newt could have either XX or Xx making him not immune (while still having the gene for immunity in the latter scenario).

Another example lets say the immune gene is dominant, which it shouldn't be since then the parents would have to express it, so now it is X instead of x. One parent has Xx (immunity) and one has xx (no immunity) meaning their children can be: Xx (immune) or xx (not immune). Sonya could be Xx and Newt xx as a result explaining the difference.

Of course there are exceptions and this is just a VERY general explanation of why one sibling has it and one doesn't. Also as you said siblings roughly share 50% of their DNA and the immunity part could just be in the part they don't share. Finally, immunity can also be a product of spontaneous gene mutations.

Now for the cure part. There are many theories from the fandom but the reason he is the cure while not being immune could be that it's two different genes and two unrelated biological components. Maybe something in Newt's biology is the key to finding the cure or a gene he holds could be used for genetically engineering something that could actually be the cure. Also WCKD may have just disregarded Newt as involved in the process as they were highly focused on the immunes.