I guess you mean RippleLabs... yes, the client code is just hosted on ripple.com for convenience, you can run your own rippled to connect to and set up a webserver with the client locally too (it's what I do for example), I linked the repositories before. I'd recommend to use the tagged release versions and not bleeding edge code as it is under active development.
So if I set up a centralized webserver as you suggest, the trades will be distributed with the Ripple Labs hosted transactions?
rippled is not a webserver, you would probably deploy ripple-client on a local webserver though. Then tell the client to connect to your local server instead of the official public servers and your local rippled server needs connectivity to other rippled servers of course (just like bitcoind). I am not sure what you mean by "Ripple Labs hosted transactions".
I meant Ripple Labs generated blockchain, although there isn't a way to validate that independently. As a black box, the Ripple Labs servers just seem to be a clearinghouse of transactions similar to Liberty Reserve.
Well, just like with Bitcoin where you kinda need to use the Satoshi generated block chain, you kinda need to use the ledger chain RippleLabs started. You are free to run your own fork of course, but this has similar consequences as with Bitcoin (you'd create an Altcoin with Bitcoin rules or an alt-ledger with Ripple rules). Their validators are widely used, but not the only ones out there, so while they are currently still quite central to the network the network could very well survive without them. You can run your own validator by the way too, some people who are critical of RippleLabs would probably add yours to their UNL to have even more diversity. I might announce my own one "soonish" too, once it has a bit better hardware and runs more stable.
The ledger chain as you call it is not mined. It is arbitrarily assembled by an algorithm subject to the whims and fancy of Coinlabs, Inc. and/or Ripple Labs. It is not an open source project and it is not even publicly auditable. Again, this centralized ledger is akin to Liberty Reserve which was shut down and the operators were arrested and indicted. It just sounds too risky to get involved with.