On how/if I should ask my supervisor about lead authorship?
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.
You should ask him about this. There's nothing rude about doing so, authorship discussions are part of academic life. Unless he's been unreasonable in the past (and it doesn't sound like he has) you can approach it as a fairly casual discussion. Other authors (including the people who did the original research) would need to agree. Make sure it's a polite discussion, not a demand!
This can differ a great deal between disciplines, but in mine I would expect you to be the first author and the professor to be last author.
Since the professor has asked you to write the paper, I presume that he does not intend to be the first author. Almost no professor will want to be the first author of a paper whose text is obviously written by a student. The first author is usually presumed to be the author of the text, i.e., the person who actually wrote the text, and very few professors would sign a substandard text as the first author.
You can have a look at the professor's publication list to determine whether he often lets his students be first authors. If he does, he is likely to do the same in your case.
There is a simple method to remove the uncertainty about the order of authors. Prepare some very initial draft of the paper: write the title, abstract, and a few paragraphs of the main text, and write the authors in the order you think is most fair. Then email this initial draft to your professor together with some question about the draft. The professor will open the draft and see the order of the authors. If he makes no objection against the order of the authors, consider it negotiated. Later you can always refer to that email and the absence of objections. It is entirely normal to include the author list in a paper draft; moreover, failing to do so is rather weird, especially if you use an article template taken from the journal website, as such templates have a special place for the author list.
Do it early, not late. I know one postdoc who wrote a very good paper, in which he put himself in the fist position and his professor in the second, and then sent it to his professor, who then made only one correction: he swapped the positions. The actual contribution by the professor was nothing but funding the postdoc's salary. The postdoc was pissed off, but simply swallowed it because he needed a prolongation of his postdoc contract and was unable to take postdoc positions in other cities and countries for family reasons. If the postdoc had known in advance that he was going to be the second author, he would have at least invested less effort in that particular paper. Furthermore, if the postdoc had used the above described method in advance (i.e., in the very initial stage of writing the paper), the professor would possibly not have swapped the positions, because he would have known that such an action would strongly demotivate the postdoc and result in a paper of a worse quality.
Please note that students generally lack writing skills, so your professor well may have to re-write the paper or invest a lot of time and effort to teaching you how to write papers. If this happens, it may be fair to put him in the first position. If he really invests a lot of time and effort and suggests putting him in the first position, don't strongly argue with him, especially as you need a recommendation letter.