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
I like the simple slogan: homotopical algebra is the nonlinear generalization
of homological algebra. Let me assume that you value and appreciate homological
algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the coarsest level homological algebra is based on the idea of resolutions, i.e. that to perform algebraic operations on objects we should describe them in terms of objects that behave well for the given operations.
Now let's observe that homological algebra is a linear theory,
in the sense that it deals with things like vector spaces, modules over a ring,
and more generally objects of abelian categories. What if your interests
involve more complicated objects that are not linear? for example, rings, algebras,
varieties, manifolds, categories etc? philosophically it still makes sense that
we have much to gain by resolving in some appropriate sense. Homotopical
algebra is the language and toolkit built for this explicit purpose, and with many explicit applications. The $\infty$-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory.
Some key examples:
$\bullet$ Hodge theory. For me (and I assume many other algebraic geometers) the first
instance of homotopical algebraic thinking I encountered was Deligne's construction
of mixed Hodge structures on the cohomology of complex algebraic varieties, one
of the most powerful tools in modern algebraic geometry. The idea is that the functor "de Rham cohomology" is very wonderfully behaved on smooth complex projective varieties, and
most importantly carries a rich extra structure, a pure Hodge structure. We can take advantage of this for say any singular projective variety if we use the idea of
resolution, in the form of a simplicial object (a convenient nonlinear version of a chain complex) --- we replace the variety by a simplicial smooth projective variety which
is equivalent in the appropriate sense, in particular will produce the same
measurement (cohomology). The existence of such is deep geometry (resolution of singularities) but its explicit applications don't require explicit knowledge of this geometry. It now follows that the singular variety's cohomology carries the appropriate derived version of a pure Hodge structure, namely a mixed Hodge structure.
$\bullet$ The tangent complex. Another seminal circa 1970 application is the
Quillen-Illusie theory of the tangent complex. Again we want to do basic geometry - this time calculus - on a singular variety, or perhaps let's say a commutative ring, so we resolve it in the sense that befits the problem. We like affine spaces for
taking derivatives etc, so if we want to calculate derivatives (tangent spaces) on a singular variety we should resolve it by such --- replace a ring by an appropriate free resolution (this time a COsimplicial variety). This gives us a way to extend the
basic tools of calculus to singular varieties, with many corresponding applications.
$\bullet$ The virtual fundamental class. This is an elaboration on the previous point which is much more recent. We would like now to integrate on a class of singular varieties,
so need a version of the fundamental class. The varieties in question arise as moduli spaces (say in Gromov-Witten or Donaldson-Thomas theories), which means they are relatively
easy to resolve in a natural way (express as a derived moduli problem). As ordinary varieties they are very badly behaved (eg are not even equidimensional) but the derived moduli problem naturally carries a fundamental class.
$\bullet$ In representation theory the key objects
of study are again nonlinear --- associative algebras (or equivalently their categories of
modules). Thus to perform algebraic operations on these algebras we gain much by
allowing ourselves to resolve them. As mentioned above the geometric Langlands program
is one place where homotopical language is extremely useful, but one can find the same issues in studying say modular representations of finite groups (eg the theory of support varieties and stable module categories). More generally Hochschild/cyclic theory, the "calculus" of associative algebras/the fundamental invariants of noncommutative geometry, are natural applications of homotopical algebra. There are many spectacular achievements in this area, one famous one being the Deligne conjecture/Kontsevich formality/deformation quantization circle of ideas. The cobordism hypothesis, in my view one of the pinnacles of homotopical algebra, has among its many facets a vast generalization of Hochschild theory.
Best Answer
An $\infty$-categorical perspective is given here https://arxiv.org/abs/1507.03913 and a triangulated expansion of those ideas is here https://arxiv.org/abs/1806.00883
More or less, perverse sheaves are the heart of a certain $t$-structure that you build "gluing along a perversity datum".