I will answer all of your questions in one shot here.

No you can not shoot anything but 22lr out of your gun in its current configuration and get it to function properly. The reason is that 22 longs and shorts will not feed properly nor will they have enough recoil to make the bolt cycle fully. 10-22's are of a blow back design which means that the force of the case against the bolt is what cycles the firearm. You could load longs and shorts one at a time, fire them, and manually cycle the bolt to eject the spent casing. This is perfectly safe to do.

A 17hmr is based on a 22WMR (22 magnum) case that is necked down to 17 caliber so the cases are too long and the bullet is smaller than the bore. However you can get a kit for the 10-22 and shoot 17 mach2 ammunitionout of it. The kit contains a 17 mach2 chambered barrel and either a stiffer recoil spring or a heavier charging handle to compensate for the added pressures of the mach 2 cartridge.