Grammar – Is ‘He Lay on the Bed Gazing up at the Ceiling’ Correct?

SHORT ANSWER:
No, the writer did not omit a relative pronoun (and linking verb) and
Yes, the sentence may be written with ... actions which are necessary ...

LONGER ANSWER:
It is true that many grammarians treat constructions of this sort as you suggest. They call this construction a reduced relative clause—a relative clause from which the relative expression and linking verb have been deleted.

I believe, however, that this is not what is actually happening here, and that your rewrite is not an expansion of the sentence but just a paraphrase of its meaning. There is another way of looking at this which does not involve ellipsis (omission) and which is (in my opinion) the way native speakers actually think of what they are doing. What is involved is a rule that an adjective with a complement must come after the noun it modifies.

For instance, the ordinary position for the adjective necessary would be before the noun it modifies, actions:

Software quality assurance is a planned and systematic pattern of all necessary actions.

In your example, however, the writer wants to expand the thought by explaining what these actions are necessary for. This is accomplished by exploiting the fact that adjectives like necessary and ready and eager behave just like verbs in one respect: they take complements, terms which ‘complete’ the sense:

I am ready. Ready for what? ... I am ready [_complementfor dinner].
She is eager. Eager to do what? ... She is eager [_complementto leave].
The actions are necessary. Necessary for what? ... The actions are necessary [_complementto provide adequate confidence.]

(And by the way: nouns can take complements, too, in the same way:

The actions provide confidence. What kind of confidence ... The actions provide confidence [_complementthat an item or product conforms to established technical requirements].)

But in English you are not permitted to put an adjective with a complement in front of the noun it modifies, because that would break the connection between the adjective and the noun:

^∗Software quality assurance is a planned and systematic pattern of all necessary to provide adequate confidence that an item or product conforms to established technical requirements actions.

Consequently, we move the adjective phrase—the adjective with its complement—after the noun, as in your example.

There is further discussion of such postposited adjective phrases here, and a discussion of ellipsis here. And you might be interested in a discussion on linguistics.SE where a poster explains that this rule, and different rules in other languages, are designed to make complicated sentences easier to interpret.

^∗ before an utterance marks it as unacceptable.

This blew my mind for a while. Indeed, a very good question. This is the example of a negative concord.

They ain't got nothing to say means They don't have anything to say.

I think it is more idiomatic than grammatical.

Along with OxfordDictionaries, such phrase is also found on some authentic sources including Google Book Result (Behind Ghetto Walls).