The main purpose of SpamCop.Net is to report spam sources, not web sites. If it reported your "website" and not an e-mail you sent, then that was merely as a convenience to you and your ISP to let you know that someone is "Spamvertizing" (also see this page) your web site. If the unlikely event that you have any control over whoever used your web site in their e-mail, insist that she or he stop using it. If not, there is nothing you can (or need to) do unless your ISP is threatening to punish you, in which case you can refer them to this Answer and the SpamCop Forum web site in general.
In answer to your question, "But I was wondering if this report is immediately sent after just 1 complaint to SpamCop or if there's a treshhold within SpamCop after which SpamCop sends this report to my hosting provider?" the answer is that each report from a SpamCop user is sent to the hosting provider, at the option of the SpamCop user (by default, it is sent). It is not really a "complaint" but, rather, a "heads-up" that your web site is being referenced in spam.
Note from originator, Steve T: thanks to "Su" for improving my original post.