Is it possible change the volume on an individual alarm?

If you're using Sleep tracking alarms, you can adjust the volume of individual alarms in your Sleep schedule in the Health app.

No, the built-in Clock doesn't allow you to set a gradually increasing volume. However, there lots of other alarm apps available in the App Store. As I recall, that's often a feature.

I'm not sure what you mean by "minimized screen". If you get a notification that the alarm is going off, put your finger on it and pull down. You can snooze or turn it off, at least as I recall.