Yes, that is normal for nitrate to go up, maybe not by that much that quickly, but it will go up for the rest of eternity.
Probably just finished cycling out the last of the build up.
You are at a point, I think, where you will start to see what kind of bioload you have on that tank. Over the next couple of weeks keep a close eye on nitrate. I try to have mine below 20ppm but I have one tank that is overloaded that gets up to 40ppm, I have to change water in it twice a week. You will see how quickly the nitrate will build up and how often you will have to change water over the next couple of weeks.