Hi,
You don't need a chain whip tool to fit a cassette, only to remove it.
Very odd the skewer doesn't fit, never seen that before. You can try to get a replacement.
You are right, you need to adjust the brakes. Leave 1 or 2 mm gap each side and enough cable to meet your preference.
Plenty of videos on YouTube on how to do it.
Well, true, I did only need it to take the cassette off, either way I had to take a second trip to the bike shop to get one, which was a little annoying. As for the skewer, it was odd. It fits, but the shape of the lever means I can't lock it in the direction I'd like. Not really an issue since I had one anyway. But I can adjust the brakes tomorrow morning and take the wheels for a spin. The red nipples look great whilst the wheel is spinning, too.