While the finish is a bit rough, it looks to be an honest Mauser Luger. It's a plus that all parts match and for your first Luger, I think you did well.
And yes, clean it as you would any other pistol. A used toothbrush will help get into those areas that are difficult.
If you plan to shoot it, I suggest that you buy a MecGar magazine. They are excellent quality and help preserve your original mag. That "take down tool" is also used to load the magazines and save your thumb.
