Maybe I'm wrong about this but has Lancer really used his noble phantasm yet? Thought they had to scream out their name in order to activate them so it seemed more that the two abilities of his lances are passive rather than active. The only question then is why they didn't work before but considering they started showing off their abilities after the bandage was taken off it's probably safe to assume they were being sealed, most likely because they are passive and he can't turn them off which in turn would mean it'd be easier for them to figure out his identity.
As for assassin no one really noticed him with the help of magic etc and Kiritsugu made a point of monitoring both that crane and Saber for a reason so if assassin shows up on that crane it'd only be a matter of time before they spotted him. Besides concealing your presence doesn't equal invisible.
I thought she didn't have Avalon since she didn't have it in the fifth war, if I remember things right Kiritsugu is the one that has it not Saber.
I'm mostly wondering if it'll heal at all since they said the spear was cursed and the wounds won't heal, if it won't then it's a really big handicap. But yeah since she mostly uses both her hands when fighting the loss of her thumb is quite big. Nor 100% on this but from when I tried kendo the left hand was more important than my right as I only used the right to aim while the power came from the left.
I think most servants would die if they got hit by the full force of any servants noble phantasm unless we're talking about Caster from the fifth war and see above I'm not entirely convinced Lancers used any of his yet.