DC Series motor and its starting
As you know (because you asked the question), series wound DC motors will continue to speed up until they self-destruct or they reach an equilibrium with their load, like @Phil mentioned. So why does this happen?

Going from statement 5 to statement 6, your logic broke down when you jumped from the present moment "low back-EMF" to calling the resulting high current an increasing trend. Yes it is true that there IS high current when it is first starting (hence using them for high startup torque applications like engine starter motors), but as you know, a second later when the motor physically speeds up, the back-EMF also increases. The increasing back-EMF REDUCES the current in the armature and so statement 8 is reversed: If = Ia, so the field actually weakens. The weaker field means the new back-EMF is lower, so now at this present speed and present current, the motor still hasn't hit its equilibrium where supply voltage = back-EMF (assuming no load), so it speeds up! And this keeps going up and up until the small remaining torque balances whatever friction and windage losses the motor is experiencing.

You ever watched two friends try to one-up each other's stories? That's like how the field and armature enable each other to reach new fascinating and literally unbelievable heights.

*An interesting note about series DC motors is that under certain circumstances, the torque is proportional to the square of the armature current unlike other motors where the field strength is fixed so the torque is directly proportional to current. Here's a good writeup on the various characteristics of series DC motors: http://www.engineeringenotes.com/electrical-engineering/electric-motors/dc-series-motors-characteristics-and-performance-electrical-engineering/36546