Matt
Welcome, a wealth of info here about the Handi. I'll try to answer your questions, but some I know I cannot. Based on your mentioned calibers and what you want, I'd suggest getting the 223
and sending it back in to get the 308 barrel added on. FYI, any centerfire caliber barrel, except the 10ga, can be added to your frame, BUT it has to go back the H&R to be fitted at the factory.
Sorry, but the rimfire barrels ONLY work on rimfire receivers. But, you get one rimfire caliber, and all the other rimfire barrels will swap out on it, just like the centerfires. There's some wholesales our local might be able to go through, but I do not know their names. I'm sure others here will.
Welcome and enjoy the handi, when you get one. Don't overlook the stainless Ultra Handi in 45-70, they are a real blast.