Is this possible to do with teams?

Oskar Kuus, Yes, this is (point 4) possible.
From power automate you need to message in the form of adaptive card in Teams. Adaptive card is json string. You can send message using HTTP action (there are many different ways available to send message). You need to pass this json string in body. To create adaptive card, check this documentation.
In Adaptive card you can collect user inputs and you can capture card submit event in power automate. Once you have the event captured, you can save all the inputs to which ever DB you wish to save it.
To capture reply on that card, you can utilize the Teams message api's which will bring all the reply for you.
Please check documentation for more information on Power apps and Power automate.

Let me know if you are stuck anywhere. All the required documentations are listed in this post.
Happy Coding :).