In his paper: A Survey of Basic Stability Theory (http://link.springer.com/article/10.1007%2FBF02760649#page-1) Makki presents two (basic) facts:
We work in a monster model of a complete stable theory $T$ .
1) Given a type $p$ over $A$, then there is exactly one extension of $p$ (upto $A$ conjugacy) to an ideal type $\bf{p}$ s.t. the set of $A$-conjugates of $\bf{p}$ has cardinality $\leq{2^{|T|}}$. The set of these types is denoted by $O_{p}$. The elements of $O_p$ are called the non-forking extensions of $p$.
A type $q\in{S(B)}$ with $A\subseteq{B}$ is called a non-forking extension of $p$ iff $q$ is the restriction of some $\bf{q}\in{O_{p}}$.
2) Any ideal type is a non-forking extension of some type over a set of size $\leq{T}$.
We say $B\overset{\vert}{\smile}_{A}C$ iff for any tuple $b$ of elements from $B$, then $tp(b/AC)$ does not fork over $tp(b/A)$.
Then he lists properties of forking: Namely, invariance, existence, monotonicity, transitivity and symmetry, in that order.
My questions are:
1) I have managed to show all the properties except for symmetry follow from the two basic facts. But I would like to know: How do you prove symmetry from just these two facts and the other four properties?
2) How do you show using the two basic facts that $T$ has to be stable?
The arguments for symmetry and stability will be rather more involved than for the other properties, and I'll just give you a reference. Chapter 12 of Casanovas' book Simple Theories and Hyperimaginaries is called Abstract Independence Relations. Casanovas gives a list of properties (satisfied by nonforking in simple theories). Makkai's basic fact (2) is one of these properties (Local Character), and the others are closely related to the properties of forking in stable theories that you've established.
Corollary 12.6 says that symmetry follows from these properties.
Theorem 12.22 says that stability of $T$ is equivalent to the existence of an independence relation satisfying these properties plus Makkai's basic fact (1).
Disclaimer: I think that symmetry and stability should follow from the basic facts and the properties you've established, but different authors state the properties of independence relations in different forms, and I haven't thought through how Makkai's statements line up with the statements in Casanovas. You may have to make some adjustments and extensions.