Good luck with your Dane search - I think there is a lot of great advice here and so I will weigh in with just a few points that I think are particularly important:
-First, I DO think a non-refundable deposit is a huge red flag. Never met a reputable breeder who required one and I would never buy from anyone who did. What is the purpose of this if not money? Reputable breeders have more homes than puppies and do not need to do this, and it sends the message that irrespective of what I have to offer, I want some earnest money up front that by the way, if I don't have what you are looking for - I am not giving back. That isn't good business no matter what business you're in - period.
-I wouldn't be concerned with not having a written application - but I would never advise you buy from someone who does not offer a contract. If in doubt, just search this board alone for lots of examples of how it has hurt people and the puppy. Contracts are not created equal - the best contracts are written clearly about the puppy - not protecting the breeder but the dog! At a minimum, the contract should spell out what is required of you in terms of care, environment, and commitment and what is required of the breeder in terms of health and reimbursement. Promising to replace a puppy with a puppy is a classic puppymill tactic so avoid. Good breeders give money back if there is something crippling or genetically devastating that occurs. And the contract should always spell out that the breeder will always take the dog back under any circumstances if you can no longer care for it....and you promise to return the dog to her under the same.
-Last, I can't emphasize the health testing enough. OFA, CERF, cardiac, thyroid - all pertinent health tests appropriate for a Dane - be sure to see them. OFA certs don't mean much if you don't look at siblings also - clear parents, grandparents, and great grandparents only provide statistical odds of 30% clear so you need to understand what each of those groups have produced (all of their offspring) to really have the best potential picture of what they are producing/carrying.
Oh, and if I can put a final plug in for rescue - I will shamelessly do it. I got a call this week from someone asking for help with their Dane - she is aggressive and they don't want to keep her. I encouraged them to return her to the local Dane club rescue where they got her and they said that "they are too unethical and abusive" and won't send her back there. As a result, I had occasion this week to do some research into the Dane clubs/rescues, and contact a few Dane breeders I know (one of which is right by you) and was frankly saddened and dismayed by what I have learned about the state of the breed. Rescues so overwhelmed with dogs they are no longer accepting them and the club in general not able to support the numbers that are out there. In my geographic region, the rescues even have a well known sleazy operation on their websites as a suggested place to send your unwanted Dane because they are over burdened. I can't imagine this since most giant breed clubs I know of have been working to shut this operation down, and yet the Dane club is turning to them and advertising them - very unfortunate and a testament to the current state of things. So always consider please....you never know the gem you can find there.
Good luck!