Every time, apparently.
But mostly they had a white thermal cover over them.
This is Neal Armstrong's Apollo 11 helmet from here.
Apparently credible but unsourced discussion here: http://www.collectspace.com/ubb/Forum14/HTML/000900.html (see "Matt T"'s comments).
Based on @OrganicMarble's answer I did some further reading. @William-Rem's comment asks:
Is that a quadrafilar antenna with... a ground plane?"
and the answer appears to be yes; the four rods pointing radially below the quadrafilar "cloverleaf" are labeled as a ground plane in Figure 6.
As @OrganicMarble already points out, the VHF antennas were used ...
It's one of the VHF antennae.
Locations shown here.
1st image from here: http://www.ninfinger.org/karld/My%20Space%20Museum/pjlmpics.htm
2nd image from here: https://www.hq.nasa.gov/alsj/a11/LM5Structures.gif
This document https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20090015392.pdf shows that VHF was used for comm between the CM and LEM, and ...
According to Quanzhi Ye (via tweet; while not a primary source, he is planetary scientist who understands Chinese) it is for simulating Mars' lower gravity while testing obstacle avoidance for the Chinese Martian landers. There is some additional information on the testing program for the lander and Long March 5 rocket, which will carry it to Mars, in this ...
They probably were part of the support and hold-down structure of the satellite stack.
You can see them in green on each side of the stack in this picture (Credit to Origin_of_mind on reddit):
They were probably jettisoned shortly before the deployment of the satellites. This also seems to be consistent with the optical characteristics (see here)
В руках у каждого из них небольшие чемоданчики - там находится система жизнеобеспечения скафандров.
In the hands of each of them are small suitcases - there is a life support system for spacesuits.
What cosmonauts do just before the flight
They're portable cooling units for the spacesuits, as you guess, but they aren't used aboard the spacecraft; instead, the suits are plugged into the spacecraft's environmental systems. They're only used between suiting up and arriving at the launch site, hence they aren't stowed aboard the Soyuz and aren't with the crew at landing time.