I think those are same. One thing I can say that it depends on the person. I hope it helps you.

I think "やって" gives a little emphasis on the "actions" to be made. So, with "どうやって切るの", you can be asking means/methods of cutting (e.g. by scissors, knife, etc), but with "どう切るの", you can be just asking manners of cutting (e.g. number of cuts, shapes etc.) For example, your hair stylist would ask you "どう切りますか？” , but not "どうやって切りますか？". Having said that, both can be used to mean exactly the same thing too as already discussed. Hope this clarifies :D

By contracted do you mean like, removing it makes it more casual? Also, there was another one. Is どうやって like "How do?" Or is it "How" But implying the verb? Same for こうやって

Let's see. どう切るの？ is a contracted form of どうやって切るの？ It is like gonna, wanna etc... So どうやって切るの? is a polite way than どう切るの？ but almost same. If you want to say it in a polite way, you can say that どの様に切るのですか？ Let me know, if you need further help.