The picking:
Usually, when a chicken is getting picked on, he or she was added to a flock after the original flock had established a pecking order.
There are other instances but a new bird inserted into an old flock, almost always, the new bird gets picked on. sometimes, if the
old flock is substantially bigger than the new bird, they will even kill it. If it is a hen, it will become more apparent when she reaches
age of egg production and the rooster starts to show her more attention. With a rooster, the picking becomes more evident when
the new rooster reaches breeding age, most of the picking, at this point, is done by the other rooster. Its all about the pecking order

...
The light:
Light is essential for egg production. If there is 11 hours light available to the chickens per day, they will continue to lay and postpone
their natural molt (basically fooling nature with the artificial light). In this case, stress builds over time... not a nervous stress but a physical
stress on the body. Chickens that are given artificial light tend to lay out a couple of years early... remember.. a molt isn't a condition
among hens.. it is a natural thing... a period of recovery... The feathers coming in (dark quills very thick and pronounced) does sound
like she is, or is at least trying to go into a molt... her avoidance of the rooster is a sign she is trying to molt, but the rooster will not allow it.
Try this... Turn the light off for about a week... they will start using the sun as a natural cycle of "daylight"...
Keep this in mind:
Picking is nother personal... it is natural dominance coupled with survival of the fittest... she is the weak one... therefore she is the one picked on...
Try to give her a separate feeding location and if you have time, pull her from the flock and give her a mix of half kitten food and half chicken food.
Up her protein to insure she is not weak and losing her feathers due to malnutrition (which is always a strong posibility if she is getting picked on
and not allowed to eat)... The more meals she misses the weaker she gets... the weaker she gets... the more vulnerable she is to illness and disease...
Honestly, if they pick on her until you see her bleeding anywhere, i would separate her and try to nurse her back to full health...
Climate controlled:
Good! they don't need an artificial "heat". The only "good" reason to provide an external heat source is in high humidity areas where the nexting area
stays damp or wet due to humidity... in which case... its not a heater but a dehumidifier... They have enough down to take care of the temperature...