For condition 2, constant and finite variance: The time series volatility around the mean (i.e., the distribution of the individual observations around the mean) does not change over time. Am I right to assume that this basically means the line of best fit does not change?

Partly yes, stationary series will be mean-reverting i.e. the line of best fit would be horizontal. But a random walk with no drift could have that and still not be stationary because it does not have constant and finite variance.

Fourcastle, How can u determine randon walk with or without drift? Without - b0=0 and b1=1 with drift - b0 not equal to 1 and b1=1 Correct or not? Also, mean reverting means b1<1, does it mean b1>1 mean diverting.