Let's label the three conditions you wrote as 1'), 2'), and 3)'. Combine 1') and 2') by saying for any finite cover by compact subsets... (I hope you follow what I mean). Also, in 3)', you better mean that $\bar U$ is compact, otherwise it doesn't make sense.

So now we're reduced to 1) and 2). Also, let's not talk about sheaves of abelian groups, but sheaves of sets. Sheaves of abelian groups are just abelian group objects in the category of sheaves of sets, so there is no harm to start with the latter. Now, if you want to say you are doing "sheaf theory" you better actually have a Grothendieck topology. Here is what you can do in your situation:

Given your space $X$, define the category $K\left(X\right)$ as the poset of of compact subsets of $X$ and their inclusions. You can define a Grothendieck pre-topology, by saying a covering family is a *finite* family of jointly surjective inclusions. Being a sheaf for this topology is equivalent to condition 1) (i.e. 1') and 2)'). Let us call this topology $J$.

Consider the category $O_c\left(X\right)$ of open subsets of $X$ which have compact closure. Since $X$ is locally compact, these form a basis for $X$. There is a Grothendieck pre-topology on this category which is the usual one (restricted to this subcategory), except we only allow *finite* covering families. Sheaves for the associated Grothendieck topology will in general not be sheaves on $X$ in the classical sense, unless $X$ is compact. However, given a sheaf $F$ on $K(X)$, we can define $F(U)$ for a $U$ in $O(X)_c$ by $\varinjlim F(K)$ running over all compacts containing $U,$ but note, this is the same as $F(\bar U),$ since the poset of compact subsets containing $U$ has $\bar U$ as a terminal object. Note, we may also just simply remark that there is a functor $$cl:O_c(X) \to K(X)$$ induced by taking closures and what we have done is defined $cl^*F$. Then, 1) implies that $cl^*F$ is a sheaf for finite open covers as well, since $X$ is locally compact. It would seem that this implies 2) follows automatically, however I haven't checked carefully.

**EDIT**: This fails in general! Condition 2) is equivalent to for all $F$ sheaves on $K(X)$, $cl_!cl^*F \cong F,$ i.e. the co-unit of the adjunction $$cl_! \dashv cl^\star$$ needs to be an iso on $Sh(K(X))$, which is if and only if $cl^{\star}$ is full and faithful when restricted to $Sh(K(X)).$ Since $cl$ is itself full and faithful, it implies that $cl_*$ is, and hence we get that $F$ must lie in the image of the pullback topos $$Sh(O_c(X)) \times_{Set^{K(X)^{op}}} Sh(K(X))$$ in $Sh(K(X))$. Concretely, this means there is a finer Grothendieck topology $J'$ on $K(X)$ which does the trick. It is possible to write down explicitly, what the covers are, but I will not attempt to do so here. To get an idea of how to do this though, look at how I construct the "compactly generated Grothendieck topology" here: http://arxiv.org/abs/0907.3925

**The following should work with either $J$ or $J'$ **

If all this is right, then all you are talking about is a sheaf on $K(X)$ with the Grothendieck topology I have mentioned. Given such a sheaf $F,$ if you restrict to a compact $C$, $F|C$ defines an ordinary sheaf on $C,$ by extending it to opens the way I described. Since every point $x$ in $X$ has a compact neighborhood, it should follow that sections are determined by their stalks, by reducing to the case of ordinary sheaves on compact spaces.

Now, if $i$ is a closed inclusion, $i:V \to X,$ then $i$ induces a functor $$i^{-1}:K(X) \to K(V),$$ by intersecting with $X.$ This functor preserves covers, so the functor $$i_*:Sh(K(V)) \to Sh(K(X)),$$ defined by $$i_*F(C)=F(C \cap V)$$ for all compact $C$ in $X,$ is well defined. It has a left adjoint $i^{\star},$ which is simply given by procompositing with $i.$

**Note: The following probably fails now, with the new topology:**

I am not sure how to get $j_!$ for *open* inclusions, since for open covers this is usually induced by the inclusion functor $O(U) \to O(X),$ but we have no inclusion functor $K(U) \to K(X).$ BUT, we do have an inclusion functor $$K(V) \to K(X);$$ it is right adjoint to the functor $i^{-1}.$ It also preserves covers, so we get an induced functor $$i_!:Sh(K(V)) \to Sh(K(X))$$ by left Kan extension. It is in fact left adjoint to $i^*$ so, from a *closed* inclusion we get both $i_*$ and $i_!.$

Cech cohomology is automatic, since it makes sense in any topos. **This is always true**