About the first, I'm not sure what you mean, you wanted to access the inventory WHILE you're shopping? what could be the use of that?
About Rei, yes but 3 levels could be right, I mean: at level 2 you get 1 skill point. Then Rei is autoleveled up to 5. So it's 5-2 = 3 levels. BUT, you would get one point at level 4, and the next one at level 6. So it's right that you get 1 skill point only. Hope I managed to explain
Levelup is automated, so I don't think there could be a bug in that (even if it appears instantly, the code running behind it is the same as you get the levels gradually, so it really should work fine).