You need to stiffen your arm but just a little. If the arm is too relaxed, it can't carry your body weight and you fall on your face.
It sounds to me like you need to push off more and go down less. Lot's of times beginners think of falling straight to the ground. This can cause your forward arm to "give". Instead of thinking of falling down, push yourself out- forward. This will make your downward progression smoother, as you can distribute the weight as you fall. Remember rolling is about rolling forward or backward, not about dropping straight down. If you throw a wheel straight down, you might damage the wheel, but if you throw it forward, it will start rolling the second it comes into contact with the ground, making the transition smoother.
Not this either. While I agree that you need less down and more forward, pushing off does not fix inherent structural issues, and often leads to them not being rectified in the long term.
When it comes to rolls and break falls, I find the methods in the following videos to be helpful. Firstly, try the rolling method in this vid:
This is how I teach rolling these days, and I like to do it as slowly as possible, without kicking off. Just allow your weight to come forward until you go over naturally. Daniel demonstrates this well in the video. When you move on to standing, try to get the same feeling, so that you bend your legs to get close to the mat and don't kick off. Notice that you will roll at an angle, not straight ahead. Let this happen. It is more natural.
The next video is more to do with break falls than rolls, but I really like the approach:
Just make sure you don't try to do it all at once, and get comfortable with the early exercises before you move on. The first exercise where he rolls in a circle is very good for getting comfortable with touching the mat. Also, the exercise where he flips off someone's back is easier when you use a soft(ish) object like a big punching bag. My wife still doesn't have the courage to do a standing forward roll, but she is happy to do a flip off a punching bag. It is surprisingly easy.
Anyway, don't try to jump into it. Try to relax and get comfortable with touching the mat. Start low and slow and build up from there.