Is it possible to install SentinelHub API locally/on-premises with my own images repository / data source? Is there any opensource version of API?
No, Sentinel Hub API cannot be installed on-premise.
Perhaps best to check the openEO:
Thank you. So, if I understand correctly, the code for both the backend of SentinelHub and Copernicus is closed, meaning it’s not possible to use it in my own project. What about Evalscript? Is it in the same situation, considering it’s a part of the backend?
You will find a repository of open source custom scripts here:
I’ve seen that repository, but it seems to contain only scripts. I’m more curious about how it works under the hood 
In the back-end it uses Sentinel Hub.