Technically any derivative is a differential equation, just a trivial one.
For instance
d(f(x))/dx=f'(x)
since first derivatives can be treated as fractions,
d(f(x))=f'(x)dx
then integrating we get f(x)=int(f'(x)dx)
But obviously DE's can come in all shapes and forms and there are even such things as partial differential equations, integral equations, and difference equations which are very closely related to differential equations. But yes and no, any solution of a differential equation is in terms of some function. This is because your given an equation that is in terms of a function's derivatives and your asked to find the function which satisfies that equation. As I showed before sometimes it is simple algebra tricks and then integration. But for most higher order differential equations there are various methods and some of them are very complicated, while some are brute force methods such as series solutions, while still some methods are very elegant. In physics the first higher order equations that you will come across are usually oscillation. As for your last question the first equation v=at+v(0) is actually the solution of one of those trivial differential equations that I showed you before where v(0) is the initial velocity, then for constant acceleration a(t)=d(v(t))/dt=a implies that d(v(t))=adt and integrating we obtain that v(t)=at+C where C is the constant of integration and evaluating at 0 we have v(0)=0+C=C which implies that v(t)=at+v(0). You are very right though, they are an extremely elegant method to solving physical problems.
