横になる, as an expression, means to lie down. Like, if you want to tell someone you are going to lie down and take a rest, that’s an expression you can use. You can’t use 水平になる for this.
横になる, as just the sum of its parts, means to become horizontal. If you were talking about something that wasn’t horizontal becoming horizontal, you could say that. You could use 水平になる for this.
横に動く would mean to move horizontally as opposed to vertically. Your position changes, not the alignment of anything. Depending on your perspective, it could actually mean a few different things. Some of those might be appropriate for 水平に動く, like a pixel moving horizontally on a screen. Others, like you walking left and right, would not. This has more to do with the multiple meanings of 横. 水平 has fewer meanings.
ロッケトは垂直に発射した I’m not sure exactly what you mean by asking how it’s different. Do you mean, can 縦 be used instead of 垂直? I think technically the answer is yes, but if you’re talking about a rocket, it sounds more appropriate to use a “heavier” kanji compound than a kunyomi word like 縦. A small child who doesn’t know a difficult word like 垂直 might say 縦, I suppose.