A1. You need to be able to set the DNS servers for the domain name to point it to the hosting server. Assuming you are hosting with a web hosting provider they will have some documentation explaining what you need to set. The domain name registrar will have documentation explaining how to to set the information for the domain. Once you point the domain to the hosting, it should fully update and be visible in 24 hours or so.
A2. The web hosting may come with email service or it may not. You should check to see what the case is. If there is no email provided, you will have to arrange for separate email hosting (such as Google Apps for Business) if you want to use firstname.lastname@example.org addresses. In either situation you will need to edit the MX records for the domain to point mail to the correct mail servers and again, everyone will have instructions for how to do this.
A3. In the short run, no. In the longer run you will want to make sure that whoever register the domain name is a primary stakeholder in the company or whatever the venture is so that when things come up for change or renewal it happens smoothly. I cannot tell you how many times I've been hired to deal with a web site that where the domain was registered by someone's buddy 10 years ago and no one knows how to login to change settings. Re-establishing control of a domain name through the registrar is usually a difficult and painful process designed to prevent domain hijacking but ultimately very frustrating for legitimate recovery attempts.