Welcome to the forum. Here is a link to a page that shows the different sensor sizes.

To answer your question, 'Is sensor size important?' I would say yes it is BUT it depends on the shooter, and their intentions. For me, I was planning on moving to a full frame sensor, but then I took a good long look at my shooting style, and decided that the APS-C size sensor was a better fit for me, and my style of shooting. It really depends on what you are planning on using your camera for. I like to get in close to wildlife, and the narrower field of view I get from my smaller sensor helps me do that.

One area where I think sensor size is very important is image quality. The more pixels you pack into a sensor, the smaller they have to become to fit. For example, if you put 10mpxls on a full frame sensor, and 10mpxls on a much smaller 4:3 sensor, you can make the pixels on the full frame sensor much larger. The larger the sensor, the better they are at capturing light. The result is cleaner images at higher ISO's. That's not to say the smaller sensors are bad though.

Something else to consider is the size of the camera. As you stated, how the camera feels in your hand is a very important thing to consider. A camera with a larger sensor, will require a larger body. A camera with a smaller sensor can result in a smaller, lighter camera.

Also, smaller sensor are cheaper to produce, and that will play a big role into the price of the camera.

So when you look at the big picture, I don't know that sensor size by itself is a huge issue, but it does play a role in just about every aspect of choosing a camera that you mentioned in your original post. Cost, how it feels in your hand (size) image quality, purpose and price.