A suggestion (what is suggested, the object of the verb suggest) is a noun. You can't use an infinitive verb where a noun is required. Going here is a gerund, a form of the verb go which functions as a noun. [Note that suggested can followed by that and a finite verb, "suggested that we go". That's different; that is a conjunction introducing a clause expressing purpose, end, aim, or desire.]
Please include the research you’ve done, or consider if your question suits our English Language Learners site better. Questions that can be answered using commonly-available references are off-topic.
Why is this sentence correct? She suggested that he go to the cinema. I would definitely use goes instead of go.
grammaticality - Why is this sentence correct? “She suggested that he ...
She suggested going for a walk. OR She suggested you go for a walk. suggest doesn't take "to". Neither do several other verbs like recommend and advise. I don't understand how you organized that page. The second group has some odd things in it. But we don't usually do homework here.
To say "I suggested he went" may mean "I supposed he went", meaning the action was performed. Hence "I suggested he go" is the correct sentence, since the action was not yet performed, and we don't even know if it would be.
tenses - I suggested we go together / I suggested we went together ...
It's suggested that he work hard. - Correct. After the use of some verbs such as suggest, demand, insist, etc., which imply an obligation, followed by "that", the verb in the object clause usually takes the subjunctive, which is "work" in this case. It may also be useful to know that (some) British speakers might not follow this rule.