As i said, cleanly stopping the sword is currently my biggest problem. To stop the sword you need to do tenuchi, here it's usually described as a movement like twisting a wet towel. But to make that work you need to hold the sword properly, which beginnerst in most cases don't. It's hard to describe without showing it, it's funny you are asking a lot of questions i recently explained someone in training

. If you look at your palm, there is an elevation due to your thumb. If you go across your hand (parallel to the wrist, right next to the wrist) there is a little valley and then another elevation. Beginners almost always hold the sword in a way that the hilt is in that valley. But that's wrong. That second elevation must be on top of the hilt. This position feels strange if you aren't used to it.
Argh, trying to simply describe it won't work. I made 2 photos:
This is the usualy beginner mistake and is WRONG:

This is CORRECT:

It's the same for both hands btw.
No, it's a forward and backwards cut, more backwards than forward. When the cut to the horizontal is finished, the hilt should be in your center and about one fist length away from your belly. It's important to be relaxed, don't completely stretch your arms, that's also a popular beginners mistake.