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Yes, in a warm zone of 7 or higher, Ginger is hardy.

Here is a link with more information.
https://www.gardeningknowhow.com/edible/herbs/ginger/growing-ginger.htm

It can be difficult to make plant id's without growing zones and more details.

The plants to the left of the pot, may be Torch Ginger.

In the 2nd image you have a Split Leaf Philodendron.

Often times neighbors may have similar plantings and can offer some help.
I would also suggest you visit local garden centers and you can often find matches to your own landscape plants. Snip a leaf or two to take with you.

I am going to assume you are growing a shell ginger plant.

Dead and damaged leaves are pruned in the spring.
In zone 9-10 these plants can grow 9feet tall.
Cutting them down will not work to control the growth and will remove the flowers, as you should prune of cut the stem after blooming. Once they bloom, that cane dies.

https://www.gardeningknowhow.com/ornamental/flowers/ornamental-ginger/growing-torch-ginger-lilies.htm

If you are using you mint for cooking remove the flowers so the plant doesn't put all its energy into developing seeds. This is the objective of plants for the survival of the species, but not always to our advantage.

Torch Ginger is not suitable for container growing, this is most likely the issue with your plants not blooming and the weak stems that your plants have.

Here are some articles with more information.

https://www.gardeningknowhow.com/ornamental/flowers/ornamental-ginger/growing-torch-ginger-lilies.htm

https://www.gardeningknowhow.com/ornamental/flowers/ornamental-ginger/butterfly-ginger-lilies.htm

Not all varieties of ginger are edible. The best (and easiest) way to make sure you can eat your ginger is to plant a root you buy from the grocery store. Edible ginger will bloom, though it won't be as showy as its ornamental cousins.

This article should give you more information:
https://www.gardeningknowhow.com/edible/herbs/ginger/growing-ginger.htm