From locality to deterministic hidden variables According to this link here Bell said:
"My own first paper on this subject ... starts with a summary of the EPR argument from locality to deterministic hidden variables. But the commentators have almost universally reported that it begins with deterministic hidden variables."
Also
"Despite my insistence that the determinism was inferred rather than assumed, you might still suspect somehow that it is a preoccupation with determinism that creates the problem." (Bell 1987, p. 150)
My understanding(keep in mind that I am not a physicist) is that the violation of Bell's inequality shows that you cannot have a theory with local realism that can explain the predictions of quantum mechanics.
My interpretation of realism is that there are some hidden variables and that the properties we are measuring are there independently of whether we measure them or not. However, the violation of the inequality means that you can still have a theory with local non-realism or a theory that is non-local but does not abandon realism (see here for more details). So you can abandon one of the two.
My question is, starting from the assumption of locality, how does Bell infer that there are hidden variables/determinism(as I showed in the quote by Bell above). It seems to me that if local non-realism is a possibility then you cannot infer local realism just from locality as Bell did. How can you say that locality also means predetermined values, since then you are ignoring the other possibility i.e. local non-realism.
 A: Following the link you provided, the Bell quote you refer to is in Section 4 ("Bell's Theorem").
(The article at that link (lets call is "SZ") is taking a particular stance which is confusing and I will come back to it below.)
The statement "...EPR argument from locality to deterministic hidden variables..." summarises EPR's argument (Bohm version): Suppose an entangled state, of two spins 1/2 particles, with total spin zero is created at the origin and the particles travel in opposite directions to detectors at A and B which are far apart.
If you detect +1/2 at A then you immediately deduce that the other particle at B will show -1/2.  Assuming locality (no way for the measurement at A to influence results at B instantly) suggested to EPR that the perfect anti-correlation between the results of A and B must be due to pre-existing values for the two spins, hence "from locality to deterministic hidden variables". 
That is, EPR used locality and the properties of a specific entangled state in Quantum Mechanics to argue for the incompleteness of Quantum Mechanics: to them hidden variables were implied.
However, EPR's conclusion was an extrapolation from the properties of a specific entangled state. So the possibility of deterministic hidden variables underlying QM is a conjecture separate from locality (see eg Ref. 1 for a long discussion).
Bell then set up a scenario which included the two assumptions of locality and hidden variables to come up with a theorem: Local hidden variable theories make predictions different from quantum mechanics. Experiments agree with quantum mechanics.
You are right: the conclusion is that you either accept locality and give up on realism (which is just standard QM), or try to keep realism and give up on locality -- which is possible in the non-relativistic case through the de Broglie-Bohm theory --- a rewriting of standard QM which had inspired Bell. 
A confusion caused by many papers, including the SZ article you quote,  is that they emphasize only one assumption of Bell-type theorems --locality --- and conclude that violation of bell's inequalities implies nonlocality. But there are always other assumptions in the derivation of Bell-type theorems (see Ref.1) so the nonlocality conclusion is unwarranted. 
Indeed the formalism of quantum theory (more explicitly quantum field theory) is manifestly local. There are no nonlocal influences, though there can be nonlocal correlations that are stronger than those in classical physics. (Yes, quantum theory is unintuitive).
Another nice reference clarifying the assumptions involved in Bell's theorem, with minimal physics input, is this paper. 
