What should I do if an elderly woman takes my arm when getting into a car, and should I ask her to let go, and is it permissible for me to offer her my hand to help her into the car if she requests it?

If a woman is old and neither desires nor is desired, there is no harm in holding her hand or arm to help her ascend or descend. In fact, this is considered a good deed, as long as temptation is absent from both sides. This is because the prohibition exists due to fear of fitna (temptation/strife), which is non-existent in this case.