How to consume an Bedrock Studio App from a end-user perspective

Hello.

how does the end-user consume the chat app?

Bedrock Studio itself is accessible as long as you have an IAM Identity Center user.
End users can gain access by passing them an IAM Identity Center user.
https://aws.amazon.com/jp/blogs/aws/build-generative-ai-applications-with-amazon-bedrock-studio-preview/
https://docs.aws.amazon.com/bedrock/latest/userguide/br-studio.html

Is there a way to share the chat-app without providing builder grants - so read-only/chat-only mode? 2.1. If not, are there APIs to build my own app UI and consume the chat-app?

To share a chat app without giving builder permissions, I think you'll need to create your own UI to be able to run Bedrock's API.
If you need your own UI, you need to create it yourself.
Bedrock Studio creates the Bedrock knowledge base behind the scenes, so in order to use it, you need to run the API using an AWS SDK (such as boto3).
https://docs.aws.amazon.com/bedrock/latest/APIReference/API_agent-runtime_RetrieveAndGenerate.html