My Google mail account groups incoming mail quite differently than the
email accounts I've used in prior years. I'm concerned that I may
receive contest entries and miss them since I have no obvious way of
removing the entries from my inbasket once I've processed them (or, at
least, I haven't yet discovered a good way).

My point is: I acknowledge every contest entry I receive. As of 1:55PM
(eastern time) on Tuesday, I'm all caught up with processing entries
and have acknowledged all I've received.
By Thursday night, if you have not received an acknowledgment for your
entry, send me a note (you can even resubmit your entry to be safe)
and I will make sure I got everyone included.
You can even wait until Friday morning but I can't guarantee that I
will acknowledge prior to noon