To answer your questions:
Yes, you should contact the program directly and you should follow the instructions on the program website. So, in your case, you should email the listed point of contact, i.e. the Graduate Administrator. You should not attempt to directly contact the admissions committee unless the website tells you to do so / provides you with the contact info. Instead, the point of contact will forward information to the right people.
As for content of the email, I think it's okay to say everything you proposed to ask. That is, you should let them know that you received the waitlist notification email from the Dean of the Graduate School and that their program is your #1 choice and you will certainly take the offer should they make you an offer.
You can ask about the waitlist information however I am not sure if asking for the specific position on the waitlist is useful. Many programs don't have ranked/numbered waitlist. Instead, after the review of applications, they might choose to select some number to make an offer, some number to go into a waitlist pool and some other number to decline admission. Then, if they need to fill spots from the waitlist pool, they might actually review the waitlist applications again and make selections from that. The committee may use information known about the interests of the students who already accepted the offer and select people from the waitlist to match department needs (e.g. if certain subfields are not well represented in the initial responses, they might favour those subfields from the waitlist). Alternatively, they might make ranked waitlists but separately for each subfield so there's no meaningful rank/position they can provide to you. Another complication is that often schools already make more offers than they have spots for (similar to how airlines "overbook" a flight) because they know a large number of their admitted students will decline. This means that some schools might not always make offers to a waitlist unless a larger than expected number of people turn down their offer, which means knowing that your #1 on the waitlist isn't that meaningful. And finally, it is possible that the school automatically puts everyone who is not made an offer onto the waitlist.
Therefore, I would advise you to ask about the waitlist in a different way that might provide more meaningful information for you. Perhaps you would considering framing the question something like: Do you have any more information on whether the department is planning to make offers from the waitlist and the timeline of these decisions?
I think this framing will provide you with more information about the waitlist directly from the department, which is more useful than the generic info you have so far from the Graduate School. Their answer might provide some insight on how they operate their waitlist. And at this point, it is likely they will say something like "mid April" for timeline of decisions but it's a good question to ask.
Finally, you should certainly let them know any other additional information about your application that you didn't include originally. Just state whatever benefits you are available for. However, keep in mind that department policies may not allow the Graduate Administrator to add info to your application package at this stage. In addition, for many schools, it's not money that limits how many students that can be admitted but time and other non-money resources (e.g. office space, etc.). This is why someone who is able to fully fund their own PhD isn't necessarily guaranteed to be admitted. However, it is still worth providing this update to the department---they can decide what they will do with it.
I don't know if you have heard back from other programs yet. If you have more than one standing offer right now, it is probably a good idea to consider existing offers carefully and pick one of those current offers to be the one you'd commit to if your top choice does not work out. Then, you can decline the other offers you have. If you haven't received more than 1 offer yet, you should do this as offers come to you in March. Just keep one offer in hand at all times, declining the ones you don't favour as much once you are able to make that determination (e.g. after a visit or a Skype chat or something). This will help waitlists at other places move along. You would hope others who currently hold an offer at your top choice would do the same. It might even benefit you since if someone who is holding an offer at your top choice is actually waitlisted at a school where you hold an offer, if you decline yours in favour of another school, then it might open up a spot for this other person who can then decline your top choice and potentially open up a spot for you! Or you can probably imagine this in much more complexity between some large number of schools. So if people are responsibly declining offers or withdrawing applications as soon as they are certain they are no longer interested, it will be better for everyone.