Unexpected behaviour of Nothing inside List inside Association

The observed puzzle is resolved by the following two observations:

From Leonid's answer in the linked q/a:

Association is HoldAllComplete. Once it is created, its parts will then normally be held unevaluated.

From Nothing>> Details

Nothing is removed as part of the standard evaluation process. It is not removed in expressions that are held or inactive.

You can Map ReplaceAll on the association to force removal of Nothings:

ReplaceAll[d -> Nothing] /@ assoc (* or *)
Map[# /. d -> Nothing &]@<|a -> {c, d, e}|>

<|a -> {c, e}|>

Use Replace with level spec All rather than ReplaceAll:

Replace[<|a -> {c, d, e}|>, d -> Nothing, All]

<|a -> {c, e}|>

Related: Is there a difference between Replace with parameter "All" and ReplaceAll

Pattern matching to a function evaluation inside an Association answers your question about bug vs expected result.

This will not do what your input suggests but let me anticipate your needs:

<|a -> {b, c, d}, e :> {d, Print[1], d}, f -> d|> //. 
  {a___, d, b___} :> {a, b} /. (*this way Print is not evaluated*)
  d -> Nothing

<|a -> {b, c}, e :> {Print[1]}, f -> Nothing|>

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