1. He decided that she should go shopping.
2. His decision was that she should go shopping.
3. He made a decision that she should go shopping.
4. He suggested that the party be put off.
5. His suggestion was that the party should be put off.
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Can we use all the sentences? Can we omit 'should' as well? Sentences 1 through 3 are the same. Sentences 4 through 6 are the same. Am I right?
6. He made a suggestion that the party should be put off.
Yes, you are correct that sentences 1 through 3 convey the same meaning. They all express the notion that he decided or made a decision that she should go shopping.
Similarly, sentences 4 and 5 are equivalent in meaning. They both suggest that he recommended or proposed that the party should be put off.
Regarding your last sentence, sentence 6 is also correct and conveys the same meaning as sentences 4 and 5. It states that he made a suggestion that the party should be put off.
In terms of your question about omitting 'should', it depends on the context and the intended meaning. In these particular sentences, 'should' is used to express the speaker's opinion or recommendation. If you omit 'should', the sentences may become more factual statements rather than expressing a suggestion or proposal. However, it would still be understandable without 'should'.