"Fraïssé limits" without amalgamation

This question has been around for a long time, and the OP himself has known the answer for almost as long! But just in the interest of putting some references out there, I'll give an answer:

Let $\mathbb{K}$ be a class of finite structures. Then $Gen(\mathbb{K})$ contains a unique isomorphism class $M_{\mathbb{K}}$ (often called the generic limit of $\mathbb{K}$) if and only if $\mathbb{K}$ is countable up to isomorphism, has the JEP, and has the weak amalgamation property:

(WAP) For any $A\in \mathbb{K}$, there exists $B\in \mathbb{K}$ and an embedding $f\colon A\to B$ such that for all $C_1,C_2\in \mathbb{K}$ and embeddings $g_1\colon B\to C_1$ and $g_2\colon B\to C_2$, there exists $D\in \mathbb{K}$ and embeddings $h_1\colon C_1\to D$ and $h_2\colon C_2\to D$ such that $h_1\circ g_1\circ f = h_2\circ g_2\circ f$: $\require{AMScd}$ \begin{CD} A @>f>> B\\ @VfVV @VVg_1V\\ B @. C_1\\ @Vg_2VV @Vh_1VV\\ C_2 @>h_2>> D \end{CD} Note that we don't require $h_1\circ g_1 = h_2\circ g_2$, so the images of $B$ in $D$ under $h_1\circ g_1$ and $h_2\circ g_2$ may be different.

The usual amalgamation property (AP) has the same statement, but with $B = A$ and $f = \text{id}_A$.

References: The equivalence between WAP and the existence of a generic limit was identified by Ivanov in his paper Generic expansions of $\omega$-categorical structures and semantics of generalized quantifiers and independently by Kechris and Rosendal in their paper Turbulence, amalgamation and generic automorphisms of homogeneous structures. A clear presentation (in the wider context of finitely generated structures) can be found in the recent paper Games on finitely generated structures by Krawczyk and Kubiś. I also gave an exposition (in the wider context of classes of finite structures with specified "strong embeddings" between them) in my PhD thesis (Section 4.2).

So that answers the first and second questions. For the third question, a structure $M$ is the generic limit of its age if and only if it satisfies weak homogeneity: For any substructure $A\leq M$, there is an embedding $f\colon A\to B$ (with $B\in Age(M)$) such that for any embedding $g\colon B\to C$ (with $C\in Age(M)$), $C$ embeds in $M$ over $A$, i.e. there is an embedding $h\colon C\to M$ such that $h\circ g\circ f$ is the inclusion of $A$ into $M$.

The condition of weak homogeneity is implicitly there in Ivanov's original paper (since it's what powers the back-and-forth argument establishing uniqueness of the generic limit), but he doesn't give it a name. Again, my thesis is a reference for this equivalence.

Kubis wrote a lot on this subject.

This.

This.

I also have a small contribution on this subject.

There is also an even more abstract point of view, started by Rosicky in '80.

The general motto is that the amalgamation property has a key role in the construction of the Fraissé limit, any weakening of it corresponds to a weakening of the universal property limit.