'Will you please help me?' (main verb=help)
'Yes, I would be happy to help you.' (infinitive as the complement of the adjective 'happy')
Is this what you mean in your question, Q1: I would like someone to explain to me when we use to help and when we use help in the sentence. Is there any rule that I should know? ?
Yes, you are correct. It should be, 'There are five people in the room.'
I hope this is helpful,