Inductors in series with different currents
I think this is the dual of a problem that has been discussed here on CO about paralleling a charged and an uncharged capacitor. Try to find that question for an in-depth explanation (and remember that you must swap the non-zero series resistor in that explanation for a non-infinite parallel resistor in your problem).

For you inductor question: 


with ideal components you problem is unsolvable, because (as you noticed) there are two conflicting requirements for the current at t=0: 10A and 0A.
if you try this in real life the voltage across the switch will rise to the level that is required to let the 10A flow unimpeded. This will likely involve some interesting effects. This is a reason that switching high currents is difficult and not friendly to a switch.