It's normal to show unpublished work to colleagues, sometimes for their input/advice/suggestions for edits, sometimes because they're working on similar work (whether collaborating or not), or, like in this situation, to demonstrate work for a job/grad school application.
I agree it's a little unusual to ask directly, but only a little, and in this situation it makes sense, especially if the publication is listed on a CV as "submitted". More typically it would come up if, say, you were having a conversation with someone at a conference or after a talk. You're discussing some topic, and Person A says: "Oh we have some recently submitted work on this! If you'd like, I can send you the manuscript."
If the field allows for preprints (that is, if journals will not consider preprints to be prior publication and accept them as original articles), it would be a good idea for someone applying for positions to make sure their latest work is out as a preprint if possible and to include these details with their application. (of course there are other benefits to preprints, as well) Since you say people use arXiv in your field, then assuming the specific journal doesn't have an unusual policy against preprints, that's how I would approach this: reply with "oh of course! the paper has been uploaded to arXiv here: ".
I think it would be a courtesy to other authors to make sure they don't have any issues with sending the paper to one outside person (and I'd double check with her advisor, just in case there are any weird politics - in that case it seems that advisor is you!), though I personally wouldn't be bothered at all if the first author of a paper I contributed to chose to share it with someone else. I think other authors should definitely agree before posting a preprint with their name on it (just like they would agree to a submission), but I wouldn't expect any opposition if the work is already in condition to submit to a journal.