Flexible Nonmetallic Conduit inside wall?
Liquid Flex May be Okay

NEC 2017 356.1 Liquidtight Flexible Nonmetallic Conduit 

-- Basically says you're good but I cannot tell the size of the flex.
   It looks to be 1in. diameter, which is fine if that is the case.

-- You may need to secure every 3ft or less to be in accordance w/ 
   356.30 for securing runs longer than 6ft if that were to happen.


However, you have other issues to attend to...

Disconnect Means Missing

NEC 2017 Part II 225 Buildings or Other Structures Supplied by a
Feeder(s) or Branch Circuit(s) require a Means to disconnect all 
ungrounded conductors per 225.31

-- Basically anytime you have a separate structure fed by a branch
   circuit or feeder, you need a disconnect means AT THE STRUCTURE 
   to disconnect all ungrounded conductors (and no more than 6). 
   This is often done by installing a main-breaker type panel as 
   opposed to a main-lug which you have used as shown in your 
   pictures.

   The NEC does not define structure and as such a piece of wood 
   driven into the ground with a sub-panel on it could qualify as
   structure to the authorities having jurisdiction.


Back-feeding breaker as a "Disconnecting Means"

Per NEC 2017 408.36 (D) Back-fed breakers used as a Disconnecting Means
MUST be bolted or secured in place as to not be pulled out.

-- Basically this means you cannot just stick in a 60A breaker
   in you current panel and use it as a disconnecting means because 
   it is not bolted in place.  You could use a hold-down kit as 
   mentioned in your comment, but I think you'd do better with a 
   bigger main-breaker panel.


You still need a Disconnecting Means in the outhouse in addition to the rule of 6

Concerning your comment "...Apply 225.33 (A)"    

-- To be clear: Every structure requires at least (1) disconnecting 
   means but no more than 6 to disconnect all ungrounded conductors.

   225.31 has (4) exceptions to the location of the disconnecting
   means, but none are related to the rule of 6.        


Bushings

NEC 2017 352.46 Bushings for PVC based Connectors &amp;
NEC 2017 300.4(G) Insulated Fittings

-- If the feeder conductors are 6 AWG or larger you will need a 
   bushing for the PVC connector entering the bottom of the panel 
   box.