You can set up a phone line or P.O box with the same terms of service agreement. Twitter is not sacred. You can call that "co-opting" those services I suppose, but I don't see why that's a bad thing. Services exist to be used.
Disney is going to use some common sense when reviewing these tweets and filter out the trolls and people disputing the terms. They are not out here trying to trap helpless children into a contract, just make some marketing materials. Again, this is not the first time this has been done and is not controversial. Maybe I underestimated the number of troll tweets, but I'm sure Disney will figure it out just fine.
Yes they are. Did you not read their tweet? They literally called it a contract and said that they believe anybody who replies to them with that hashtag is agreeing to its terms. Their choice not to exercise the rights they are granted by that contract doesn't mean that there isn't a contract.
The only thing that does mean there isn't a contract is that their logic is bullshit and they can't unilaterally decide that somebody else has entered into a contract with them.
The problem is Disney co-opting two regular parts of Twitter's service, and then claiming that represents a Disney service. It isn't, and can clearly be used for purposes beyond establishing a contract.