I propose that this is what you "really" mean to say by 2): "what is the maximum length $t(L)$ such that for any $0$-dimensional scheme $Z$ of length $t(L)$, the evaluation morphism $H^0(L) \otimes \mathcal{O}_X \to L_{|{_Z}}$ is surjective". This, I think, is what you mean by letting the points come together. For an general very ample line bundle $L$, the answer is of course $t(L)=2$ (very ampleness is equivalent to $t(L) \geq 2$, and it is not hard to find examples where $t(L)=2$). 

Here is the exact terminology I think you are looking for: a line bundle is called $k$ very ample if $t(L) \geq k+1$. This is a notion due to Sommese and there is a huge amount of literature about it. A good primer on it is Ch. 5 of Göttsche's paper "A conjectural generating function for numbers of curves on surfaces". This is closely related to the notion of Seshadri constants as in Karl Schwede's answer, but it is slightly different in general. 

For example a globally generated line bundle is $0$ very ample and a very ample line bundle is $1$ very ample. 

Seeing as you were taking powers of $L$, perhaps the result you are looking for is the following: suppose $L$ is $k$ very ample, then $L^m$ is $km$ very ample. In particular, if $L$ is very ample, then the answer to 2) in general, with my interpretation of the question, is $t_0=m+1$.