Whenever I end up with the same amount of attacks as my opponent I know that I'm going down.
Why is that?
It's really like 9 times out of 10. And for some reasons those lost rounds tend to result in a lost game.
This is very frustrating.
Are there certain things that influence this or am I just still having bad luck (as typically in all games)?
Nope, when UR says the mode is no-random it really is no-random. When two cards have the same attack, the one with the lowest level wins. If both are of the same level, the one who attacked first (played courageously) wins.