Are road caliper brakes usually compatible with any model of lever/STI shifter? I have a Specialized Allez 16 (2011) road bike with Shimano 2300 STI shifters and no-name brand caliper brakes. The brakes have always felt a bit mushy despite my best efforts at adjustment/replacing pads, so I'm thinking of upgrading them, perhaps to Shimano 105s.

I am under the impression that more or less any model of road caliper brakes can be used with any kind of brake lever/STI shifter. Is this correct, or should I be worried about compatibility?
Pretty much right -- you just have to make sure that the calipers have enough reach to brake (on an Allez, pretty much anything but a long reach caliper will work). There are some threads like this with some new Shimano stuff (new Super SLR) which is technically a ymmv/you're on your own, but it should work. 

Another thing you should check when replacing the brake is that the cabling was done properly -- bad brake cable routing, for example, can make your brakes not feel good.