People file lawsuits when companies have filed for IPO because generally that leads to companies being in a "quiet period" in which they cannot effectively respond publicly, which increases the likelihood of settlement, etc. It is also extremely common for competitors to seed the press with negative story lines, taking advantage of the legally mandated quiet period to remove the filing company's ability to counter these stories.

I wonder if the lawsuit was purposefully timed near the IPO announcement, in an attempt to heap extra pressure on Spotify to settle quickly.

I can't really figure out what the merits of the case are, or get an idea who these folks are. Not a lot written about them, even though they have been around for some time.

Has spotify really been playing these major artists without permission? Or is there more to the story?

Not unusual to file for 10x what you think you might collect in the end. If there really are damages, will Spotify just give them 160 million in stock?