Hi PG,
Q1:
Im not sure if you are asking this so if not just reply again...
"Vo(t)" simply means "the output voltage as a function of time", or to put it another way, "the output voltage in the time domain".
Many times the 't' is dropped when it's obvious that we are working in the time domain so we would just have "Vo".
Other common terms like this would be:
"Vo(f)" the output voltage as a function of frequency,
"Vo(w)" the output voltage as a function of angular frequency (w=2*pi*f),
"Vo(s)" the output voltage in the complex frequency domain, or output voltage as a function of complex frequency, sometimes shortened to just "the output voltage in the frequency domain".
Another common term would be:
|Vo(s)|
or:
abs(Vo(s))
or:
ampl(Vo(s))
or:
norm(Vo(s))
where all these mean the amplitude of the signal disregarding the phase.
Q2:
The equivalent resistance across C is the parallel combination of the three resistors. The right two are in series, that makes one resistor then that is in parallel with the first resistor.
This is the resistance that is used as part of the time constant of the circuit. It happens this way because the source is considered a zero ohm impedance so effectively the two resulting resistances are in parallel.
The catch is that although they are in parallel for the time constant, they are in series for the max amplitude. Then you have the two on the right which change the amplitude once again.
So to do this the way i think you are approaching it, you have the two resistors on the right which are in series so equal 6k, then the one on the left which is 3k so the 3k and 6k are in parallel for the time constant.
For the amplitude, you have the 3k and 6k voltage divider, but then you have another voltage divider to get to the output which is the two resistors on the right the 4k and 2k which cause another voltage division.
For a charging capacitor however you dont use e^-t/RC, you use 1-e^-t/RC as the basic form of the equation. You also end up with a voltage multiplier so you get this form:
Vo(t)=Vin*A*(1-e^-t/RC)
where A is due to the series resistance voltage divider effects.