I don't want to use bot framework in pop-up or chat window. Can I use the web API I created myself. When I do a call, I want the api to request it to list the returned data. Can I select multiple data listed?
From your description you seem to want to create a Teams Message extention, specifically a search extention that will be used from the compose box.
this is indeed powered by a webservice that you will write and host,
however the communication channel between Teams and your webservice must use the bot framework for communication.
Note that you do not need to write a chatbot as such;
I like to think of it as the 'communication channel' between all the Teams clients that have your app installed , and your webservice.
High level steps:
More steps and details in the messaging extensions documentation