It seems to me that the $m$ is a red herring. Since you already assume that $\mathcal L$ is very ample, why do you need a power? I'll ignore $m$ here. 

For the questions: In case the base field is algebraically closed (1) is a direct consequence of $\mathcal L$ being very ample. That surjectivity means that you can choose a global section of $\mathcal L$ that is zero and another one that is not zero at a given point.

For (2), I don't understand what you mean by "not necessarily distinct". If the same point appears more than once, then this will certainly not be surjective. Any section will either be zero or not zero at any given point.

If the points are distinct, then this seems like a strong criterium for large $t_0$. 

Let's say that $X$ is a smooth curve. Then it seems to me that what you are asking for is essentially the same as asking that if you embed $X$ via the global sections of $\mathcal L$, then the images of that set of points is in general position. In other words the set of points that might satisfy this condition cannot even be chosen to be simply general, but they have to be consecutively chosen to be general with respect to the points already chosen.