Excellent. I have the same question. I have to set up my wife's bike with a carrier for little Molly Mouse, our Chihuahua/micro-Terrier. Carrier must have a classic style because Molly doesn't go for the carbon fiber space pod look.
looked cool, but i think i want a more canvas style carrier.
My wife's bike is currently fitted with a basket having that same mounting system, which is really crappy, IMO. The clamps do not attach securely to the handlebars (basket is constantly drooping) and the mount attachment to the basket is just silly. It has all kinds of play, but you can't tighten it down because it crushes the basket. Really dumb. There must be a better way ... like a rack mounted basket.
1974 Montomery Wards Open Road, 1971 Schwinn Suburban, 2010 Surly Long Haul Trucker
There are also several strap on carriers available. The one I use with my Pomeranian is the 'Legs Out' chest carrier by Outward Hound. Think of one of those baby carrier type chest packs, modified for a small dog. Not for everyone, or every dog.
mid-80's Bianchi, converted to single speed, purple
so I bought that Pet Gear carrier, it seems to droop a bit and hit the front wheel. Do I need to get a front rack? I have a quick release front wheel, how does a front rack fit on a quick release front wheel?
Most racks attach to an eyelet on the dropout. If that doesn't work, you can use a rubberized strap of steel that wraps around the fork, to which you can bolt the rack. If you have canti/V brake posts, some racks attach to those. BTW, if your rack was designed to attach to the axle and you were intent on using it, you could have your front wheel re-built with a slightly longer axle so that QR still works.
It depends on the weight of your dog. I carry my 14 lb terrier in a front basket, but the basket has to be very solid or the "load" moves around a lot and affects handling in a bad way. A 6 lb dog wouldn't be much of an issue. Also, I built a "liner" for the basket out of thin plastic, and put a towel on top of that. Dog seems pretty happy.
You also don't mention if your carrying system has to be removable, or can be permanent. If permanent, a Wald front basket can be found to attach to almost any bike and the versions with front struts are very solid. They don't even take that long to install/remove, but i wouldn't want to be doing it multiple times a day or anything.