How to show that e.g. $\cos(z)$ is analytic using Cauchy- Riemann differential equations?

Start by rewriting: if $z$ is complex, then let $z=x+iy$. Then we have the function $\cos(x+iy)$. Now you can expand that with the rule $\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$. (Because you'll be left with bits like e.g. $\cos(iy)$, you'll may want to replace these with hyperbolic functions with e.g. $\cos(ip)=\cosh(p)$ and a similar relationship for $\sin$.) You'll be left with some complex function which we'll call $u+vi$ - i.e. let $u$ be the real part and $v$ the imaginary part. It is these $u$ and $v$ you are differentiating in the Cauchy-Riemann equations.

Writing $$z=x+iy\,\,,\,\,x,y\in\Bbb R\Longrightarrow \cos z=\frac{e^{iz}+e^{-iz}}{2}=\frac{\cos x(e^y+e^{-y})-i\sin x(e^y-e^{-y})}{2}=$$ $$\cos x\cosh y-i\sin x\sinh y=u+iv$$

And now you can check the Cauchy-Riemann equations directly, for example: $$u_x=-\sin x\cosh y=v_y\,\,,\,etc.$$

I will address your more general questions first. I'm restricting discussion to complex functions of one complex variable, of course.

Holomorphic and analytic functions are the same thing. A full proof is given in any complex analysis book, but I will give the outline. Assume all functions are defined in an open connected domain. Call functions with power series expansions at every point in their domain analytic, and call functions that complex-diferentiable holomorphic. If a function is analytic, we can expand it as a power series at every point, and elementary theory of power series shows they are complex-differentiable, so all analytic functions are holomorphic. To show all holomorphic functions are analytic, one uses a result called Cauchy's Integral Theorem to explicitly produce the required power series expansions at every point.

It is easily proved (see any book) that all holomorphic (complex-differentiable) functions satisfy the C-R equations, even without showing that holomorphic and analytic functions are the same. However, not all functions satisfying the Cauchy-Riemann equations are analytic (holomorphic). The standard additional condition is to require is that the function have continuous first partial derivatives (when considered as a function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ ) in addition to satisfying the C-R equations.

Now, showing $\cos(z)$ is analytic requires knowing how you defined it. I like to define it in terms of exponentials or power series, and in either case, analyticity is trivial. However, if you are restricted to the fact that $\cos(z)$ is just some function that satisfies the standard trigonometric properties, then I would go with the approach of Erik Pan, also posted in the answers section. Briefly: rewrite $\cos(x+iy)$ using the cosine addition formula, rewrite the result as $u+iv$ using hyperbolic functions, where $u$ and $v$ are real-valued, and then verify the C-R equations by differentiating directly. Remember to check that the first partial derivatives are continuous.

It sounds like you would benefit greatly from a good complex analysis book. A quick treatment (that covers your questions in greater detail than this answer) is available for free here.