TITLE: Categories with every indecomposable object being uniserial
QUESTION [6 upvotes]: Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. 
Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.
Questions:


Is the endomorphism ring of a uniserial object local?
Assume $A$ has the property that every indecomposable object is uniserial. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?
Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras. 
Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?
Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

REPLY [3 votes]: Every finite length indecomposable object in an abelian category has local endomorphism ring (the proof is exactly the same as for modules), and a uniserial object must be indecomposable, so the answer to question 1 is “yes”.
Here’s an example that addresses some of the other questions.
Let $A$ be the category of representations, with finite total dimension, of the infinite quiver
$$\dots\to\bullet\to\bullet\to\bullet$$
Then every object is a representation of some finite subquiver, and so every indecomposable object is uniserial.
There are enough projective objects, but no nonzero injectives, so not every projective embeds in a projective-injective, answering question 2 (even the alternative version in comments).
Also, $A$ has global dimension equal to one, answering the last part of question 4.