I know that batteries get charged evenly because of the balancer, hence we use balancer.My question was: Should I use balancer while battery pack is connected to the robot, or balancer is used only while charging?
The answer to that is somewhat intrinsic to what I wrote.
And then the second post question was: If battery pack gets unbalanced while charging without balancer, can it get unbalanced while discharging without balancer?
If the pack is unbalanced when you start discharging it, it doesn't get unbalanced, it IS
It will only get worse each charge and discharge until proper balancing is used at every charge (or it's converted to a fireball, whatever comes first.
Stop playing with lithiums and get a proper charger, it's much safer and judging from some of your posts, I am not too convinced your experience is enough that you could make a safe charger, even if you used a dedicated lithium charger chip.
Grind your fangs on something like NiMH instead, it's much safer and the experience you build along the way will be worth it.