I have a collection of several pics that I want to turn into NFT. What I am worried about is that someone can save my NFT as a file, and then make a new NFT out of it and sell it in several copies. Is that a real danger, and how can one protect from it?
You are right about this being a "real danger". This is completely possible and there is almost nothing you can do to prevent this. I am not an expert but the main thing you can do is "advertisement". You use the power of everything to publicly announce that "you are actually creating some piece of art and selling it as NFT" and then you point to your NFT collection in a marketplace (Opensea, rarible, etc).
Some marketplaces (like Superrare) verify the artists and only sell items from verified artists. This ensures that the buyer is actually buying a valuable piece of art and not a scam. But still: there is nothing prevent someone to download your image and try to sell it on Opensea. Would someone buy the scammer's NFT? I doubt... But what if he/she makes more advertisements?
Update: I forgot to mention about the "verified account/collection" of Opensea. "Sane" buyers would try to buy NFTs that are verified. Read more about here.