per-charge cost to charge a battery bank at $0.10/kwhr
1: Yes.

2: How can we possibly know whether the battery bank you describe is indeed what you have!?

3: No.

First, it helps to use proper engineering units.  Too many digits makes things confusing, as apparently happened here.  "10400 mAh" is 10.4 Ah in grown-up units.  Assuming ideal battery and power supply just to get a starting point, that's 10.4 A for one hour to charge.  (10.4 A)(3.8 V) = 40 W.  That times 1 hour = 40 Wh.  You pay $.10 per kWh, or $0.0001 per Wh.  That times the 40 Wh you will use costs $0.004, or 4/10 of one cent.

In reality charging such a battery will take substantially more energy than what you get by assuming everything is ideal.  Figuring 1 cent ($.01) should be conservative enough.