Ok, first thing to understand is that IPFS will not download any data unless you ask for that data. So if you don't want to download some content, don't open the link containing that information. It is similar to loading a website in that case. If you don't want to load a webpage, don't type the url into your browser.
If you load the content it will stay on your node for a while. Or until you 'garbage collect'. If you want to store content for longer on your node, you should 'pin' it. This will keep it from being removed.

Due to the nature of content adressing it is hard to know what is 'illicit' material in advance. There are no white/blacklists that I know of.

I can answer your first question though:

1. There is a maximum storage you can set for IPFS. It can be set in .ipfs/config under StorageMax.