the .222 and .223 are different in just a couple of ways. the .223 case is larger therefore it holds more powder which means it is a more powerful little round by a few hundred fps in comparison to the .222. So you will get more range with the .223 and more energy. They both use the .22 caliber bullet though and are usually similar in bullet weight as well. the .223 is capable of deer hunting but they sometimes require a perfect behind the should shot to be effective for a quick kill, also a harder bullet should be used for penetrationof both lungs because a varmint bullet just wont hit both lungs very well.

To answer your second question, the .17 hmr will not fit in the .22lr because the .17 hmr is a neck down .22 mag case witha .17 caliber bullet in it. So the .17 hmr will chamber in a .22 mag but not recomended. the .17 mach 2 is the necked down version of the .22 lr and will chamebr in a .22 lr but it is also not recomended either. So please dont try shooting either .17 through either .22 rifle because there is a high chance of injury in try it.