The rings satisfying your condition (for right modules) are the **right pure semisimple rings**. There are many equivalent conditions. You can find a lot of information in Section 4.5 of the book

<cite authors="Prest, Mike">_Prest, Mike_, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk). xxviii, 769&nbsp;p. (2009). [ZBL1205.16002](https://zbmath.org/?q=an:1205.16002).</cite>

or you might find it easier to access the older paper

<cite authors="Prest, Mike">_Prest, Mike_, [**Rings of finite representation type and modules of finite Morley rank**](http://dx.doi.org/10.1016/0021-8693(84)90080-2), J. Algebra 88, 502-533 (1984). [ZBL0538.16025](https://zbmath.org/?q=an:0538.16025).</cite>

As you say, such a ring must be right artinian.

It is known that a ring is both left and right pure semisimple if and only if it has finite representation type (i.e., every module is a direct sum of indecomposable modules, and there are finitely many isomorphism types of indecomposable module), which is a left/right symmetric condition. And there is a longstanding conjecture about a strengthening of this.

**Pure Semisimplicity Conjecture:** A right pure semisimple ring has finite representation type.

Or equivalently this says that pure semisimplicity should be a left/right symmetric condition. There are many positive results for particular classes of rings.

In Question $2$ you say that you are particularly interested in the hereditary case. This doesn't make the conjecture easier, as Herzog proved that if there is a counterexample then there is a hereditary counterexample. Combining this with a result of Simson, it turns out that to prove the conjecture it would be enough to prove that a right pure semisimple hereditary ring is left artinian.

There is a lot of work on rings of finite representation type, especially hereditary ones.  The fundamental result is [Gabriel's theorem][1] classifying the finite dimensional algebras over an algebraically closed field with finite representation type as those Morita equivalent to path algebras of quivers whose underlying graph is a disjoint union of simply laced Dynkin diagrams. There are many generalizations; one for general hereditary artinian rings is

<cite authors="Dowbor, P.; Ringel, Claus Michael; Simson, D.">_Dowbor, P.; Ringel, Claus Michael; Simson, D._, Hereditary Artinian rings of finite representation type, Representation theory II, Proc. 2nd int. Conf., Ottawa 1979, Lect. Notes Math. 832, 232-241 (1980). [ZBL0455.16013](https://zbmath.org/?q=an:0455.16013).</cite>
  [1]: https://en.wikipedia.org/wiki/Gabriel%27s_theorem