On how/if I should ask my supervisor about lead authorship? Some time last year, I was casually talking to my professor when he asked me to look at this paper that he and his previous student had worked on but couldn't finish. I was hesitant but I took a look and decided to work on it. I've been at it for a few months now and I've developed it beyond the scope and reach of the initial paper. As such, my prof asked me to write a new paper from scratch, but I don't know if I'll get first(or even 2nd) author on it or if I'll have to settle for 3rd behind the initial authors. 

Is it appropriate to ask such a question? I feel like I deserve it but I'm not sure if it's rude to ask for it or about it. He's going to be one of my recommendations for grad school in the very near future so I don't want to do something rude and lose any goodwill.
Yes, it is appropriate to ask, but it may not be best to demand. But personalities can get in the way of ethics, as many posts here show. "I worked pretty hard on this and having first authorship would give me a big career boost." That could work. Much better than simply demanding it as a right, even when it might be considered so. 

But, having some publication, no matter where you appear, is better than none. And don't think of this as do or die. There will be other papers and your professor's good opinion of you is worth more than first authorship on an early paper. 

But yes, it is appropriate to ask. It won't be rude unless you make it rude. 



Ideally these things are worked out before the research begins, but it often isn't. And the  power imbalance between professor and student gets in the way as does ego.