A320 Selected vs Managed [climb/descent] SOP's
 

Speaking from a practical standpoint. Do companies prohibit using open climb/descent outside of needing to meet a constraint. In other words, if Joe Q pilot wants to use open climb or descent on EVERY SID/STAR, do companies generally prohibit this? I'm asking because as I like to describe it to my students. Open climb/descent is a "dumb" way to climb/descent. Dumb, as in [open descent] the airplane goes, Thrust idle, pitch for speed.

Not prohibited at all.
Not dumb either. Simpler, yes. You can plan your descent on your own. From cruise, push the meters button, this displays the altitude in meters on the SD. Descent when distance to go equals altitude (in meters)/100.
Adjust for wind (10% if wind accounts for 10% of your ground speed), you're good to go. Monitor during the descent with altitude/slope = distance

Nothing is “dumb”, nothing is “smart”. The word You need to consider is “appropriate”. Golden Rule N^2: use the appropriate level of automation at all times.

Managed Modes are generally speaking appropriate in the vast majority of situation -But- some situations do require selected modes or a combination of managed/selected such as short term intervention to disregard an Alt or Speed constraint, descent profile management when expecting shortcuts/vectors, better descent profile optimization for non DPO equipped airplanes and so on. Use all the resources the airplane has built in, and get the maximum benefit.

No restriction on which modes you can use. If you want to make that SID restriction using VS, that’s fine too.

Thanks for the clarification..

No restriction you fly the plane how you want to open/managed/vs

Also many companies including my own, prefer you set first stop altitude in the FCU, rather than the final.

Do you ever have ATC asking why you don't have the cleared level set?

The first stop is the first cleared level, and no!

In Europe anyway, there aren’t that many stepped SIDs, though they do exist.

what is a "first stop altitude"

???

The Sid might have a final cleared level of say 8000ft but an initially you have to stop at 4000ft until passing a specified point before continuing the climb. If the 8000ft was set in the FCU and climb mode used the aircraft would automatically make the constraint at 4000 and then continue the climb to 8000. However if 4000 is set in the FCU then the pilot will have to initiate the remaining climb to 8000. It’s a more belt and braces way of avoiding a level bust, because if another mode is engaged for example vs then the 4000 constraint will be missed! Simple really.