Hey @brenner100
Thank you for the questions!
1 - You probably won't ever run a full node client because it would require you to re-index the entire bitcoin blockchain, which in my experiance takes more than 12 hours and generally is very not user friendly. It is not impossible, but it is non-ideal.
2 - Counterwallet is a blockchain.info style wallet, which is to say it's an open source terminal that lets you use your passphrase to access your tokens on the bitcoin blockchain. If Counterwallet went away, someone could install instance of counterwallet and the passwords of each individual would still work.
It is our intention to run a LTB branded wallet at some point in the near future, and for the last month we've actually offerred to the Counterparty folks to host a few backup instances just in case. They don't want us to do that yet because it would increase the number of places they have to patch when they improve counterwallet, which is an almost daily process right now (an hour of downtime yesterday morning to do a server upgrade for example)
I have always hated web wallets, but in this case the usability, built in exchange, and simplicity of getting set up (we've had 250 or so people properly sign up so far with a counterwallet address, imagine how many fewer people that would have been if we asked everyone to re-index)