Clutch lever and engine oil Kawasaki ZX-6R 636C

Hello together. I am proud owner (since the day before yesterday) of a 636c 🙂 My clutch lever wobbles very much. Approx. 2-3 cm down and back. Can I just tweak it when I turn back to the rotation mechanism for the clutch rope again? Or do I adjust my clutch grinding point with it? Is there any other way to attach it if the has so much play, or does a new one have to come? I have seen there is such a kind of “bolt” when you move the clutch lever. I can push the lever down 2cm? Would have 2 questions directly, the machine currently has 29,000 km on the clock. During the last oil change Motul 5100 was used. Unfortunately there is nothing else with it. I don’t know the oil at all, can I use it without hesitation again? Or should I rather take another at the mileage ? And 10W40 or ? At least 10W40 from the car is a term for me …. Does that come into my motorcycle too ? Thank you very much.