Traveler's checks seem like a rip off to me. If you are getting American $, you are likley also paying a significant fee. That's one of the best reason's to use a cc - you get teh transfer ate the bank uses- solong as they don't charge any extra fees. In the USA, you're best deal is often to go through a credit Union, or a cc that's got minimal fees for such transactions.
The room safes as I recall, do not have a key. I don't ever recall seeing a key. How they work is that you set the password yourself, (I think) every time you lock it.
We've never had any problems, but then, I don't recall having anything stolen from our room - even stuff that wasn't in a safe. (though I know that's pure anecdote.) My thought though is that there MUST be a way for WDW to open them in the event someone forgets a password.
The only thefts I've heard about, minimally on the DIS, are stroller thefts in the parks. Like someone stealing or accidentally walking of with the wrong stroller. (often tehre are many stroller taht look nearly identical) Or staff moving them and guests not being able to find them. I've heard about the potential of items being taken from the stroller basket.
I agree with calling your cc company. Another option is using your bank card. Ther are lots of ATM's at WDW. Just check with your bank. Tell them what you plan and pick one that has low international fees. Again, this is likely a safer option than carrying lots of cash, and you'll get the best rates.
Also, my habit for travel is to make sure we have TWO credit cards when we travel. I rarely use one of the cc, but it's a travel back up.
Beyond that, a family member once had trouble witha cc. It was a major card, and the company offered to overnight her a card to her WDW room. She didn't need it that urgently, so she just had them send it to her home intead, but if you have a major cc it can be done.
Last, for the cash you do carry, a family member of mine came up with a neat idea. He hated the moeny belts, etc. he was also unableto find travel pants he liked witha built in pocket. So he took one of his favorite pairs of pants to tailor - and had the tailor add custom 'hidden pocket' inside his pants. Just don't use it for anything that airport security won't like!