Definitely rent a car in Reykjavik (it's a small city anyway, no real point in going to Iceland if you're going to just stay there); small truck if you plan on going on route 0 (the 'touristic' road that loops around the entire island), big truck if you plan on venturing into the main lands (the paths are super fucked though, you might find yourself on awful terrain, risky cliffs, having to drive through lakes etc. so be careful with insurance). A few years back I had a short window of time in Iceland with my girlfriend at the time so we drove from Reykjavik to Vatnajökull through the southern part of route 0 and it was amazing, plenty of shit to check out on the way from waterfalls to small villages to beaches with colors and sceneries that completely change every 30 minutes and pretty much no one else on the road (depending on when you're going I guess). In Reykjavik try out that place where they serve soup in a bowl of bread (then you get to eat the bread), and C Is For Cookie is a chill café to bring your girl too.
In France try going to Paris and Bordeaux.
Denmark, I couldn't say much as I've only been to Copenhagen once, but it's super close to Malmö.