Herbs That Require Frost... Tarragon, Chives, etc.

I purchased herbs from you to grow indoors (I have no yard, balcony, or porch). I know that the chives and tarragon need a cold rest, but I'm not sure about the others. I have tarragon, chives, Italian parsley, Arp rosemary, Vietnamese coriander, African blue basil, sage, Kaliteri oregano, and Cuban mint.

Only the two herbs you mentioned need frost. Parsley must not have any, or it will bolt to seed and forget about producing any leaves - and then die. The African basil should be kept quite warm, the rest will do better if kept at about 5 to 10 degrees Celsius over the winter.

Since I purchased the herbs mid-late October and installed some fluorescent lights, the tarragon decided to grow off season, so it's sprouted a few new plants. I'm not sure if I should winter it now anyway or if I should wait until the end of winter, like Feb or Mar (I'm in central Illinois, USA).

I think if you froze it now it would freeze to death, because it thought spring had arrived.The plants may start to look weak later on and at that time I would slowly cool them down with a few nights in the fridge and then put them outside when you expect only frost, but nothing too far below freezing. Remember that the roots in a pot will get much colder than they would for a plant in the ground.

Do you water the herbs when they're in cold rest (if it's not below freezing)?

Only enough to keep them barely moist. Too much water will rot them.