"It is in effect on the date it is signed until it expires." - it will expire in the future, and that will then be the present; it has not expired now, when we are reporting.
"I want everyone's name." - as in, "I want John's name, and I want Jane's name - in fact, I want everyone's name." - each person has only one name.
"The most important thing to me is friends." - the verb 'to be' agrees in number with the subject, not the object.
"The most important things to me are friends AND SOMETHING ELSE."
"Let's do introductions" OR "Let's do an introduction" - This is an imperative; the subordinate clause does not need to agree with it. You need the apostrophe because you are abbreviating "let us".
"Five dollars is expensive." - "Five dollars" is a composite subject forming a singular PRICE. The subject is therefore singular. But you would say "I received ten dollars, but five dollars are damaged." because the dollars themselves are then the subject, not the composite "five dollars".
"They have to pay those 50 cents" AND "They have to pay 50 cents" - "Cents" is the plural of "Cent", not "Centses"
"Good friends are hard to find" - "Friends" is a plural subject, so obviously the verb "to be" must match in number.
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