Power consumed by light bulbs in series, parallel?
You're right in thinking that current and voltage varies according to time in an alternating current circuit. However, when analysing circuits with alternating current and simple resistive loads, it is much easier to work with average power instead of instantaneous power. You can use RMS voltages and RMS currents to simplify your analysis.

The 120V voltage rating you get out of a wall outlet is given in RMS. You can multiply this number by \$\sqrt2\$ to obtain the amplitude of the sine wave.

Your circuit analysis does seem right on the most part. Assuming your input voltage remains constant, adding resistances in parallel decreases overall resistance and implies increased power. Adding resistances in series increases overall resistances and implies decreased overall power.

If you rearrange for the voltage, you can see how the resistance \$R\$ of the circuit affects the power consumption of the circuit.

\$P = \frac{V_{rms}^2}{R} = \frac{120^2}{R} = \frac{1.44\times10^4}{R}W\$

Lastly, there's a few reasons why you might not observe the lightbulb to flicker when current changes direction. One of which is that a light bulb doesn't dim instantly when power goes to zero (try switching off the lamp and you'll see the filament fades away slowly) so it still produces light as the instantaneous power momentarily crosses over zero. Furthermore, 50/60Hz is generally fast enough that the flickering is indiscernible by humans.