What's the correct sequence to start thinking about a BJT circuit
I simulated your circuit in LTspice using a BC547B. Here is the result:-

 

Before t = 0 the transistor's C-B and C-E parasitic capacitances are charged to 5V, and the B-E parasitic capacitance is uncharged. 

When the 1V step is applied the B-E junction is effectively short-circuited by its capacitance, so all terminals jump up by ~0.33V as Rb and Rc||Re divide the voltage step by 3. As the capacitors charge so the Collector voltage falls back to 5V and the B-E voltage difference widens. Vbe is 'small' so no transistor action is occurring, and the effect is the same as if the circuit just consisted of resistors and capacitors. 

This continues until ~30ns when Vbe gets to ~0.55V and the Base starts to draw significant current. The transistor then amplifies that current at the Collector, pulling Vc down. As more current flows from Collector to Emitter the downward trend of Ve is reversed, eventually stabilizing at 0.39V with Vbe at 0.61V and Vc at 4.61V.

A simplified analysis could ignore all this and just assume that Vbe will shortly become ~0.6V. Then Ve would be 1-0.6 = 0.4V, Ic would be 1k*0.4 = 0.4mA, and Vc would be 5 - (0.4*1k) = 4.6V.