[RPG] Does Barkskin cast before Wild Shaping apply to your beast form

No, the polymorphed creature can only use actions that a normal creature with the new form would be able to do. It comes directly from the definition of “game statistics”. The Statistics section of the The Monsters Manual starts with:

A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.

The subsections include not only static numbers “Ability Scores” and “Speed” but also “Actions” meaning that RAW actions are also stats (and I cannot name a reason why it wouldn't be RAI as well). Under “Actions” it is written:

When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action, as described in the Player's Handbook.

Thus the polymorphed character's actions are limited to those the creature would be able to do were it a normal creature.

As for the possible inconsistency with the concentration ("A Wizard can polymorph self, therefore polymorph doesn't break concentration. If concentration is a class feature and it survives polymorph, all other class features can be used too.") there are two choices here:

1. Concentration is not something everyone has automatically. E.g. if a creature would not be normally able to cast spells it cannot concentrate on them either. In this case there is no consistent RAW answer: "statistics" cover all the aspects of creature -> if the creature cannot concentrate, polymorph breaks concentration -> polymorph cannot be used at self, even though PHB clearly states that it was intended to be used this way. Also Sage Advice says that concentration is not broken by polymorph, so this is definitely not a path to go with.

   You can’t cast spells while you’re transformed by polymorph, but nothing in the spell prevents you from concentrating on a spell that you previously cast before being transformed.

2. Concentration is something everyone can do and/or let's use common sense. The point of the offensive use of Polymorph is to render a creature useless in battle, i.e. if a creature is turned into a sheep it is as harmless as one. If the class features were preserved we would have sheep going into rage (barbarian), teleporting (shadow monk) or... turning into dinosaurs (unless druids are considered to be shapechangers and thus are automatically saved) which is bizarre. Therefore, all the class/race features are gone. As for the possible inconsistency with concentration, it comes from a simple fact that designers cannot see all the connections between all the bits of the rules and sometimes we can only rely on how we think things are supposed to work.

The errata for the PHB answers this question definitively:

If a Wild Magic effect is a spell, it’s too wild to be affected by Metamagic. If it normally requires concentration, it doesn’t require concentration in this case; the spell lasts for its full duration.

So no concentration is required, and the spell will last for its full duration.