1. The mayor ordered that the man (should) leave the city early.

Question

2. He suggested that the party (should) be put off.
3. He insisted that she (should) make a sppech.
4. They demanded that he should resign.
4-1. They demanded that he resign.
4-2. They demanded that he not resign.
4-3. They demanded he not be resign.
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Are they all grammatical? In American English, is 'should' omitted? Can we omit 'that'? Is #4-3 correct? (Thank you for yoiur help.)

Answer 1

4-1 and 4-2 are correct although 4-2 is unlikely.
4. and 4-3 are incorrect.

Answer 2

Damon is correct.

Sentences 1-4 include dependent clauses beginning with "that" which have verbs in the subjunctive mood. The auxiliary verb "should" is not to be included.
http://grammar.ccc.commnet.edu/grammar/verbs.htm#mood