We usually have our laptop with us and no, unless you don't put anything else in the safe, it does not fit. We usually have it siting on the kitchen table (charging)
Mine won't fit in the safe. Usually I put my laptop in the carrier, out of sight, but not really hidden. If you are worried, put it in your suitcase and lock it.
Yes, all of the onsite DVC resorts have broadband access. Cost is $9.95 per 24 hour period. The safes at BCV are large enough for a laptop- but then little else will fit. At OKW, the safes are not large enough for a laptop. I also leave mine out and have never had any concern about it.
I don't know the cost for the few WiFi hot spots. There is no wireless access in any of the rooms themselves.