Suggested answer:
(1) Example of proposed that followed by a verb in the subjunctive mood:
... it proposed that the Equal Pay Act be extended to cover indirect...
Example of demanded that followed by a verb in the subjunctive mood:
...he was past listening and only demanded that she phone the family solicitor...
Example of requested that followed by a verb in the subjunctive mood:
One evening the boy requested that the father tell him a bedtime story.

(2) The subjunctive mood is often used in contexts that express requests or commands. This is particularly common in subordinate that-clauses introduced by verbs such as propose, demand, and request.

(3) If not followed by a verb in the subjunctive mood, the three expressions commonly make use of a modal verb (often should) to express a meaning similar to that of the subjunctive.
Examples from the corpus:
The letter politely requested that after the initial photo call William should be left alone...
We now demanded that he should escort us into Bahdu.
She proposed that he should quit Parks and Gardens before he was laid off...

Also other constructions with modal verbs have been used, e.g.:
...the Grand Trunk proposed that it would build from North Bay...
...it is requested that the Holy One will do the honour...
...the sophisticated equipment demanded that all higher management and maintenance personnel had to be...

There are also examples where the expressions are followed by a verb in the indicative mood, although this does not happen very often, e.g.:
...it is proposed that the Minister, in consultation, with cabinet, is able to...

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Read about the subjunctive mood here
Terminology: subjunctive, indicative, that-clause