Research Fellow on Paper They Did Not Contribute- Solicited my Mentor
Putting an asterisk by his name and mine saying we share first co-authorship. 


That alone, on a draft shared for comments, would have been enough to anger anyone. In my opinion this RF is trying to carve a space in the paper by discrediting the quality of your work, which is unprofessional. The quote is also irrespective of any actual problems with your work - assuming the RFs suggestions have some merit - and any self-respecting academic would flat out refuse, unless it comes with an extensive research plan.

The behaviour you describe is so crude that actually improves your position. However, the personality of your mentor and your relationship with the mentor does weigh in. If you have a good working relationship with him, the issue can be resolved with a polite, honest and most importantly calm chat. The fact that he shared the draft without you knowing is not problematic, in my opinion, as academics do so very often. The liberties taken by the RF, however, may hide something more than a request for feedback - they go beyond a bad report. You cannot know, and you should not jump to conclusions, but you should be cautious, suspicious even. 

Some arguments that are generally valid, but for them to work you would need to stand as a professional academic, not a student:

1) Adding a co-author dilutes everyone's, but particularly your, contribution. Both you and the data-editing colleague can argue that you are not willing to add a 4th co-author (field specific), and for an entry-level academic first authorhip is very important and you are not willing to share.

2) The paper is mostly finalised and the suggestions are not substantial enough to warrant co-authorship, plus a debate on the most factually wrong comments (irrelevant self-citations etc). There is not enough work left and you are not willing to consider a change in direction due to time constraints and the distraction it will cause you.

3) Brush off presentation comments by saying that they will be corrected in the finalised version and insist that polishing up does not warrant a co-authorship, especially first authorship (similar to (2))

4) (very context dependent) Express some surprise on the nature of some comments and point out to their relevance. If you want to be cheeky, ask why the RF would be interested in participating in a paper he himself criticised so harshly.

It is important for you to be the one that frames any discussion as discussing feedback on a draft and you have not even considered co-authorship. If you are presented with the idea, e.g. in an off-hand comment, just say "no". If pressed further, politely yet firmly keep refusing. This is where your mentor must take a position and either drop the issue or force you to accept a co-author. If the latter happens, I see two options.

The first is to withdraw your participation in the draft. You are perfectly entitled to do so and maintain agency of your work, but I understand this is difficult from someone in your position and may cause further frictions and issues. The second is to accept (grudgingly), but by stating very clearly and in a non-negotiable manner that the RF will be responsible for any additional work and you will not be adding anything more to the paper, especially results/ experiments. Even if you receive some drafts or emails, you can politely repeat your stance. This is confrontational and also not desirable, but at least you will not be exploited.