Yeah, probably... I think he'd be surprised if Allen went over to the Noah, but we don't know if that's even going to happen anyway. Allen is a bit of a predictable guy most of the time...
I have a better opinion about Cross then you :D He has a calculative mind (supposedly) that is capable of predicting Allen's actions. After all, the core of Allens personality is still the same as it was when Cross adopted and educated him so I do not see anything that could prevent Cross from predicting Allens actions. Also, General does not seem to be troubled by anything as he would be if noticed Allen changing for the worst, so I guess his opinion about his student haven't changed. He still trusts him.
Yeah, he was going to be tried for heresy on the suspicion of associating with Noah. Komui asked for more evidence, so now he's just being watched. The moment they find anything suspicious, they'd probably go after him for that, so Allen can't stay in the Order. We'd never see the Inspector preventing him from fighting because by that time he'd want Allen dead completely - not just locked up in jail where the Noah could rescue him.
I think that he allows Allen to fight right now to test him, to see if Allen will betray them or not. The reason for exectuting Allen was him supposedly being Noah so Inspector wants to see, if it is true or not. While Allen is fighting for them using Innocence there is no reason to kill him.