I usually do not use the anonymous mode in Limesurvey, so take this with a grain of salt.
Afaik, in anonymous mode IP addresses won't be stored, no matter what you set in the IP address setting, but you should test this on your own by activating your survey and do some test runs to see what is actually saved. But I will try to answer your questions about IP addresses anyway.
1) What’s the difference between not saving IP-address and anonymizing the IP-address?
If you anonymize the IP-address Limesurvey will cut of the last 3 (I think) digits of the IP address. So you still have some information regarding the IP address, you at least could research the block of IP addresses, etc. So in your case, if you want it totally anonymous, you should NOT save the IP address. But as I said, I think if you choose anonymous mode you shouldn't be able to save the IP address anyway. Or a dummy IP address is saved. But test it yourself. Takes you 5 minutes.
2) What’s the point/advantage of registering or saving the IP-address and/or the referrer URL if my survey is supposed to be anonymous?
None, actually you shouldn't save it (and in my opinion you can't if you use anonymous mode). Saving the IP address and referrer can be relevant in many cases, but it would undermine your clame of anonymity.
3) Is my survey anonymous at all if I chose to save the IP-addresses or save the referrer URL? (I guess not)?Looking forward to hear from somebody more experienced than me.
If you save the IP address and referrer URL in your survey in any way it is probably not anonymous in a technical way anymore. However, I thinks far more important than the technical anonymity (there is always some way around it anyway) it "ethical" anonymity. As a professional market researcher I guarantee my respondents anonymity, despite the technical possibility to connect personal data with response data. However, based on the ESOMAR code of conduct I guarantee that we treat personal data separate from research data, so we won't analyze the information together with any personal data. Could we connect them both? Yes. Do we do it? No.
However, often you need to have at least some connection between personal identifiers and survey data, for example for quality control (e.g. in field research where we call a certain percentage of respondents to make sure their response is real, that they were treated well by the interviewer, that the interviewer was professioanl, etc.), to be able to invite them to a follow up survey, to be able to provide them with a "thank you" or prize, etc. Never the less, data is never connected to personal identifiers during analysis, which is what counts I think.
I know, some people want (or have to) gurantee 100% "technical" anonymity. Then you will have to go down the route of the "anonymous" mode with all its advantages and disadvantages.
But the issue there is, that if someone is malicous, they can always find a way around the technical anonymity if they really want to (e.g. pass on personal data via the URL and hide it in hidden question in the survey, use javascript and trackers to get to the IP address, etc.).
