Not in case of security, but you get likely more trust from your visitors. Also might include some assurance thing you can use or advertise with. But I wouldn't say it has any bad influence if it's missing, as long as you aren't a payment provider or something like that.
Generally not unless you're in a business segment where trust is more important (e.g, financial services).
Depends on your audience. Do most know what it means? Do most even notice it?