I got a seedling from a nursery in Ontario, Canada, and I was thinking it was dead (dried up only one thick stem) but I still kept it. Roots look okay but nothing will grow out of it since the last 2 months I have it. I have a light next to it to make sure it has at least 6-8 hours of light and I water it as soon I feel the potting mix is dry. I would like to know if I should keep it or get a new one in summertime.
Try to find out if that brown stick is actually dead. Scrape the outer portion of the stem to see if there is any green under the surface. If it's all brown, the stem is dead. If it's dead, go ahead and cut it off.
Your plant may well be dormant. Why don't you try putting it into a dark, cool room, and water it only enough to keep the soil from getting scratchy dry. Bring it out in the spring, after it's had a winter sleep, water it well and put it into the light. It might will sprout again. This article has more information: https://www.gardeningknowhow.com/edible/herbs/tarragon/growing-french-tarragon.htm
My stems bend and it goes over the edge of the pot. I was wondering if anyone could suggest to use something to keep it up and not bend over the edge. I was going to put a trellis to keep it going up. I live in the rainforest in Hilo. Would that be something I should try?
Anything you can work out that will hold the stems upright is fine. A trellis might work, or 3 or 4 long stakes or wires, with some yarn, tape, or wire strung between - let your creativity run wild.
You might want to put your tarragon in higher light. The stems might need to be stronger. This article has general information on growing herbs in pots: https://www.gardeningknowhow.com/edible/herbs/hgen/growing-herbs-in-containers.htm
The garden centers here never have tarragon plants. Can they be purchased via Internet?
Yes, they can be purchased on the internet. Amazon is one such place where you can find them: https://www.amazon.com/Hardy-French-Tarragon-Plant-Artemisia/dp/B0019SWI02
For more information on growing french tarragon, please visit the following link:
https://www.gardeningknowhow.com/edible/herbs/tarragon/growing-french-tarragon.htm
Happy gardening!
Should I leave my young tarragon plant's flower?
Remove the flowers to promote better growth.
Allowing the plant to go to seed will effect the plants flavor and your herbs may self seed everywhere if left to flower.
Here is a link with more information.
https://www.gardeningknowhow.com/edible/herbs/tarragon/growing-french-tarragon.htm
I have a nice tarragon plant in a pot. I can harvest the leafy stems, but what should I do with the potted plant over the winter? I live in SW Missouri.
It would be nice if you could place your tarragon in a greenhouse or protected environment for the winter, but if you can't, make sure there is mulch around the base of the plant and that the pot is wrapped or otherwise protected from freezing.
Here's a helpful article: https://www.gardeningknowhow.com/edible/herbs/hgen/how-to-take-care-of-container-herbs-during-cold-weather.htm
I overwintered my Tarragon in a cold dark cellar and it has started to grow already. Because there is no light the shoots are pale. Should I bring it into the kitchen where it will be warm but a bit dry or put it outside in my mini greenhouse which is unheated?
I would put it in the kitchen, which you may have already done. They are only half hardy and can't take a chill. You can grow it year-round indoors. Here's more:
https://www.gardeningknowhow.com/edible/herbs/tarragon/growing-tarragon-indoors.htm
There seem to be as many dead leaves as live ones. Plant is 12” (30cm) tall and growing in a 1 gallon (4 L) plastic container.
It looks like an infection. They do not tolerate moist soils. Here are some articles that will help:
https://www.gardeningknowhow.com/edible/herbs/tarragon/growing-tarragon-indoors.htm
https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm