How/whether to update PhD admissions committee about new scholarship I applied to 4 US PhD programs and I've received 3 rejections and 1 waitlist.

I also applied for a scholarship for $25,000/year x4years and just found out that I got it.

I'm thinking that I should update the school I'm waitlisted at because the scholarship could probably help my chances of getting in. I have some questions I'm hoping I can receive some input on here:


How should I go about updating the waitlist school about this scholarship? Should I do this in writing or by phone?  
Should I inform the schools that have already rejected me about the scholarship--is it possible they may change their decisions based on this new information? And if so, how should I approach telling them?
The scholarship would require me to research a certain (widely researched) disease and maintain one of my MSc supervisors (at a different institution) as a co-supervisor during my PhD. Is it likely that the school(s) will mind this (especially the second part)?


Thanks in advance for any and all feedback!
You should definitely notify the program for which you are on the waitlist, since this could matter for their decision.  If they have contact instructions on their webpage or in the notification you received about being on the waitlist, then you should follow those instructions.  Otherwise, you could try replying to the notification e-mail.  I'd recommend doing this in writing, rather than by phone, so that you can provide details.  You can just say something like "I'd like to update you on a development in my application: I was just notified that I have received the X fellowship, which ... [give the financial details and any restrictions].  Please let me know if you would like any additional information."

It can't hurt to ask the schools that have rejected you, just in case, but you should expect the answer to be no.  I'd phrase it very tentatively (along the lines of "I understand that this is probably not possible, but I don't suppose that my application could be reconsidered in light of this fellowship?").  Don't spend much time on this, since I'd expect that the most likely outcome is that someone will spend 15 seconds typing a "Sorry, that isn't possible" email in response, but I don't think anyone will be offended if you don't sound presumptuous.

Regarding the restrictions, it's hard to predict how much the department will care.  You should definitely mention these details in your e-mail, so they can make an informed decision.