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You can place the Bittersweet in a water container in a cool location. The berries will start to open as the plant dries.
This method will likely make using the stems for arrangements easier.

You can also hang the Bittersweet to dry, but they will open and become more brittle.

https://www.gardeningknowhow.com/ornamental/vines/bittersweet/growing-bittersweet-vines.htm
The female plants produce the berries and it is impossible to tell the plants apart until the vines produce berries.
Reputable garden centers will label and sell the male and female plants.

There is no way to determine what a vine is, unless the vine produces berries.
A reputable garden center will label the plants.

https://www.gardeningknowhow.com/ornamental/vines/bittersweet/growing-bittersweet-vines.htm

It is best to transplant your seedling around the time of the last frost date in your area, which should be around mid-April for you. Yes, you should harden it off as described here:

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/how-to-harden-off-your-seedlings.htm

American bittersweet is quite cold-hardy, and you can plant it in the early spring even before the last frost date, as long as the ground is thawed. Now should be a good time.

Here is how to choose a good site for them:
https://www.gardeningknowhow.com/ornamental/vines/bittersweet/growing-bittersweet-vines.htm

Unless you purchased both a male and female plant; they should be labeled by the nursery; there is no way to know what you have.

It can take years for a vine to establish and produce berries. I'm in a cold climate and just have my first crop of berries, 9 years in!

https://www.gardeningknowhow.com/ornamental/vines/bittersweet/growing-bittersweet-vines.htm

A soil test will help you determine the nitrogen levels in your soil. In general excessive nitrogen inhibits flowering.

Your County Extension Office can help with a soil test.

https://www.gardeningknowhow.com/extension-search/