Can one use channel markers to set channel-specific bad epochs in continuous recordings?

Yes, or start by creating a new group of events including the tag "bad_" at the beginning or "_bad" at the end of the event name.

When averaging trials or continuous files, if one has 4 files with good segments for a given time window, and one file with a bad segment for the same time window (for a few channels only, which are turned to bad channels for that segment/ window), the denominator of the average of the bad channel activity is 4, or 5? In other words, when averaging, the bad channel time series are ignored, or turned 0? (I believe the former, but please confirm).

The bad channels are ignored, not set to zero.
When averaging 10 files, and let's say channel A1 is bad in one trial only: it would compute the average of 10 values (sum of 10 signals divided by 10) for all the channels except for A1 which would average only 9 values (sum of 9 signals divided by 9).

Be careful with this individual marking of bad channels per trial:
It is ok in the number of bad trials is similar for all the channels, but you should avoid having important imbalance between channels: if a channel is good only in 1 or 2 trials instead of 10 for the other channels, you should mark it as bad everywhere instead (either by marking it as bad in the continuous file, or by right-clicking on the group of trials in the database > Good/bad channels > Mark some channels as bad). Otherwise you would have one very noisy signal (with much lower SNR) in the middle of your much smoother averages.