First time being asked to review a conference paper
Yes. I don't know for conference paper, but with journal, they usually end the year with a thank you to all reviewers. This "thank you" is available online and is a proof you indeed reviewed. But I doubt that anyone will ever verify this.
I think it might be unusual, but it is a good thing. First, it means your undergraduate advisor think that you can do it (abilities but also he/she thinks you know this subject). Second, lots of people are talking about reviewer overload. Using qualified PhD student can reduce that overload.
Experience, increased self-confidence and yes, recognition from you ex-advisor...


I suggest you consult this great article to help you start:

Seals, D. R., &amp; Tanaka, H. (2000). Manuscript peer review: A helpful checklist for students and novice referees. Adv Physiol Educ, 22, 52-58.

In my experience, doing my first review, I was a bit stressed but mostly excited. I wanted so much to do everything well. Thus, the authors got an honest and very complete review. If you try to be as thorough as you would like your own work to be revise, all should go well. Good luck.