I don’t think it will happen to you. I also wear glasses without nose bridge, but it rarely slides down from my nose. Once the frame size you choose is suitable to your face, no matter it has nose bridge or not, it can bring comfortable wearing experience.
Certainly not, it depends on the nose pad of your frame. If the nose pad clings to your nose, it won’t slide down from your nose; but you should often check your nose pad for avoiding its loosing and slide down.