(1) Let me first correct something. You wrote:
By Fraisse Theorem, any Fraisse class has a generec model $\mathcal{M}_{\mathcal{C}}$ such that it is homogeneous and its theory is $\omega$-categorical.
This is not quite true - your definition of Fraïssé class is too general to prove this theorem. Suppose $\mathcal{C}$ is a class of finite $\mathcal{L}$-structures with HP, JEP, and AP. Then:
In order for a countable Fraïssé limit to exist, you also need to assume that $\mathcal{C}$ is countable up to isomorphism. So I would also include this condition in the definition of Fraïssé class. (This condition is automatically satisfied if the language $\mathcal{L}$ is finite.)
Under the conditions above, it's true that $\mathcal{C}$ has a countable Fraïssé limit $\mathcal{M}_{\mathcal{C}}$, which is unique up to isomorphism. But you can't prove that its complete theory $\text{Th}(\mathcal{M}_{\mathcal{C}})$ is $\omega$-categorical without additionally assuming one more thing abut $\mathcal{C}$: for every natural number $n$, there are only finitely many structures in $\mathcal{C}$ up to isomorphism that admit a generating set of size $n$. (This condition is automatically satisfied if the language $\mathcal{L}$ is finite and relational - no function or constant symbols. So Fraïssé's theorem is often stated in the special case of finite relational languages. It is also satisfied if $\mathcal{L}$ is finite and $\mathcal{C}$ is uniformly locally finite, meaning that for any natural number $n$, there is a bound $f(n)$ such that any structure in $\mathcal{C}$ which admits a generating set of size $n$ has size at most $f(n)$. This applies, for example, to the class of vector spaces over a fixed finite field $\mathbb{F}_q$: a structure generated by $n$ elements has size at most $f(n) = q^n$.)
Now whenever $\text{Th}(\mathcal{M}_{\mathcal{C}})$ exists and is $\omega$-categorical, this theory also eliminates quantifiers (I gave a proof in my answer here). For example, in the book by Hodges, he proves that if $\mathcal{C}$ is a uniformly locally finite Fraïssé class in a finite language, then $\text{Th}(\mathcal{M}_{\mathcal{C}})$ is $\omega$-categorical and eliminates quantifiers.
On the other hand, there are Fraïssé classes with Fraïssé limits whose theories are not $\omega$-categorical and do not eliminate quantifiers. For examples in an infinite relational language and in a finite language with function symbols, see my answer here (the Fraïssé classes are the classes of finite substructures of the ultrahomogeneous structures described in my answer).
(2) Another possible misconception to correct: If $\mathcal{C}$ is a Fraïssé class, the statements "$\mathcal{C}$ has quantifier elimination" and "$\text{Th}(\mathcal{M}_\mathcal{C})$ has quantifier elimination" are totally different. It is extremely unusual for a class of finite structures to have quantifier elimination. So I don't really see the connection between questions (1) and (2). You probably should have asked them as separate questions.
If you're interested in a class $\mathcal{C}$ of $\mathcal{L}$-structures which is not the class of models of some first-order theory $T$, then proving quantifier elimination is not going to be easy! The main tool for understanding formulas up to equivalence is the compactness theorem, and for an arbitrary class $\mathcal{C}$, the compactness theorem doesn't apply. I think the best approach would be to try to use Ehrenfeucht-Fraïssé games.
(3) This question honestly mystifies me. Of course you'll have to use specific features of a theory to prove it has quantifier elimination!
There are many general tests for quantifier elimination, most of which are stated as equivalences ($T$ eliminates quantifiers if and only if ...), so if you're given a theory $T$ with quantifier elimination, you can in principle use any of these tests to verify it. Now of course for any given theory, some tests will be easier to apply to it than others. So you'll see many different approaches to proving quantifier elimination applied to different theories in the literature and in textbooks. But if anything, it's good to have lots of different approaches you can pick from!
For the first question, the point is that there are basic algebraic/combinatorial phenomena which can only occur in infinite structures. The simplest example is an injective non-surjective function: in the language with a unary function symbol $f$, the sentence $$[\forall x, y(x\not=y\implies f(x)\not=f(y))]\wedge[\exists x\forall y(f(y)\not=x)]$$ does have models but does not have any finite models. Another easy example is linear orders without endpoints: in the language with a binary relation symbol $R$, the sentence asserting that $R$ is a linear order with no greatest (or least, if you prefer) element has only infinite models.
(There are also surprisingly deep examples, e.g. Wedderburn's little theorem says that the sentence saying "the universe is a non-commutative division ring" has only infinite models.)
For the second question, this is just a direct application of the theorem. Let $\mathbb{K}$ be the class of all finite structures (in some fixed language). If $\mathbb{K}$ were $EC_\Delta$, then there would be some set of sentences $\Gamma$ such that $\mathbb{K}$ is exactly the class of all models of $\Gamma$. But:
Do you see why such a $\Gamma$ must have arbitrarily large finite models? (HINT: how large can elements of $\mathbb{K}$ be?)
Do you see why such a $\Gamma$ cannot have infinite models? (HINT: how large can elements of $\mathbb{K}$ not be?)
This gives a contradiction, so our assumption that $\mathbb{K}$ was $EC_\Delta$ must have been wrong.
Best Answer
No, we cannot. For example, take $\mathcal{C}$ to be the class of all finite structures in our language, let $\varphi_n$ be the sentence asserting that there exist at least $n$ elements in the universe - that is, $\varphi_n$ is the sentence $$\exists x_1,..., x_n(\bigwedge_{1\le i<j\le n}\neg x_i=x_j),$$ which is clearly first-order - and let $\Sigma=\{\varphi_n:n\in\mathbb{N}\}$. Every finite subset of $\Sigma$ is satisfied in all sufficiently large finite structures, but $\Sigma$ itself is only satisfied in infinite structures.
Note that we can do a similar trick with larger cardinalities, assuming a large enough language. For any infinite cardinal $\kappa$, let $C_\kappa$ be the language consisting of $\kappa$-many distinct constant symbols $c_\eta$ ($\eta<\kappa$), take $\Sigma=\{\neg c_\eta=c_\theta: \eta<\theta<\kappa\}$, and take $\mathcal{C}$ to be the class of all structures of cardinality $<\kappa$. Then every subset of $\Sigma$ with cardinality $<\kappa$ (not just every finite subset) is satisfied in some element of $\mathcal{C}$, but no element of $\mathcal{C}$ satisfies all of $\Sigma$.