“Can my fiance come to visit me in Europe while K-1 visa is processing?”
Of course, yes. In fact, you’re encouraged to visit as much as possible. For a US citizen visiting Europe, depending on the country, you likely won’t need a visa. Of course, this excludes countries like Ukraine and Russia, but most of Western Europe doesn’t require a visa for Americans visiting.
During your visits, keep receipts, take photos, and document your time together. This will all be helpful to your case in the future. It’s usually harder for anyone visiting the US while a pending K1 or immigrant visa is pending, including being turned away at the port of entry. For this reason, the US citizen going to visit the European alien in this instance, is the most common situation.