Motivated by this question I want to clarify something regarding vector/line subbundles.
Question 1 Suppose we have a morphism of vector bundles over a curve/Riemann surface $X$, $f : E \rightarrow F$. When is the image $\text{im}(f)$ a vector bundle? Is it always a subbundle of $F$?
I know that for kernels one needs to have constant rank (which let's assume is not included in the definition of vector bundle morphism for this post). Does something similar happens? I can intuitively see that wherever $x \in X$ $f$ vanishes, $\text{im}(f)$ will "lose rank", and I see that as undeserible for vector subbundles.
Question 2 Is it true, as stated in the comment of the cited question, that the only subbundle of a line bundle is either itself or the zero bundle? I can see by some vanishing theorem that $H^0(X, L_2 \otimes L_1^*)$ vanishes whenever $\deg L_2 < \deg L_1$, but... I can see no further. Could there be line subbundles of a given one with less degree or does it follow from the injectivity that they must be isomorphic (when nonzero) just by purely linear-algebraic reasons $L_{1,x} \subset L_{2,x} \implies L_{1,x} = L_{2,x}~~\forall x \in X$?
I know these might seem extremely obvious, but I sincerely cannot wrap my head around these concepts when identifying vector bundles and the associated sheaves of local sections.