I know the following is standard in differential topology, but I have barely no background in the latter and it is my first contact with these concepts. My motivation is the study of classical mechanics.
Let $M = \mathbb{R}^{2n}$ be our phase space. A vector field on the phase space is a function $X: M \to M$. A flow of a vector field $X$ is a function $\phi_{X}: \mathbb{R}\times M \to M$ satisfying: $$\frac{d}{dt}\phi_{X}(t,m) = X(\phi_{X}(t,m)) \tag{1}\label{1}$$ with initial condition $\phi_{X}(0,m) = m$.
Regarding these concepts, I have two questions.
Question 1: In the literature, one usually finds that these flows satisfy the properties $\phi(t,\phi(s,m)) = \phi(t+s,m)$ and $\phi(-t,m) = \phi^{-1}(t,m)$. I have no idea of how to prove these properties, tho. Are these consequences of the above definition (\ref{1}), or do I need something else?
Question 2: Another thing that is not clear to me is how the flow relates with the exponential of a vector field. More concretely, for each fixed $t \in \mathbb{R}$, the function $\phi(t,\cdot)$ is supposed to be given by the exponential $\phi(t,\cdot) = e^{iX}$. What does this mean, exactly? Is it $\phi(t,m) = e^{iX(m)}$? But what does a exponential of a vector field mean? And in what sense it is the solution of (\ref{1})?
Let's keep it simple. A vector field $X$ is a linear map acting on some smooth functions $f \in C^{\infty}(M)$. Then, one of those functions, let's call it $y$, forms a curve parametrized by $t$, whose equation of motion is generated by $X$, i.e. $\dot{y}(t) = Xy(t)$.
This equation of motion can be solved in the same manner as in a linear algebra context, hence $y(t) = e^{tX}y_0$, where $y_0 = y(0)$ is an initial condition. Actually, there is nothing surprising here, since $X$ is a linear map and $C^{\infty}(M)$ is a vectorial space, so that $X$ behaves as a matrix.
Now, the flow generated by $X$, namely $\phi_X(t,y(t_0))$, is nothing else than the map that makes $y$ evolve from time $t_0$ to time $t_0+t$, hence $\phi_X(t,y(t_0)) = e^{tX}y(t_0)$. Note that $m = y(t_0)$ with your notation. The two properties you mention in your first question are then straighforward; one has for instance : $$ \phi_X(t,\phi_X(s,y(t_0))) = e^{tX}\phi_X(s,y(t_0)) = e^{tX}e^{sX}y(t_0) = e^{(t+s)X}y(t_0) = \phi_X(t+s,y(t_0)) $$