1.Should backcountry skiing be aloud at resorts?(give reasons)
yes, because the ski resort does not take responsibility when you go out of bounds, therefor accessing backcountry from a ski resort is the same as accessing it from anywhere else.
2.If you have equipment such as probe,shovel,beacon should you be aloud to duck ropes?
isn't this the same question as above? if the rope just represents a ski resort boundary than yes, but if the rope is there to protect something (animal, forest, etc) than no.
3.Should everyone be aloud to do this or only people with proper training?
although you really shouldn't be in the backcountry without the proper gear and knowledge, it is impossible to regulate who is allowed to go into the backcountry because in all reality the backcountry is almost always public land and therefor, who would have the right to say no?
4.If you get caught out of bounds should there be a fine to be paid?
if you are out of bounds then you are not in bounds, which means that all that stupid shit about safety and liability at the ski resort ("know the code" stuff) doesn't apply because you are not on their property, therefor they have no right to fine you for not being on their property.
5.Would you get proper BC trainging if a mountain put it on and it wasnt extremly expensive?
in all reality you should not be going into the backcountry without the training. In Canada you can get your RAC (now called the AST 1) for (usually) around $175. I personally think that that's a reasonable price.