integration/ E2E testing with an API test-automation

I have a system that consists two parts:

• engine: reads input from an external source, transforms it and writes results to the DB
• web: exposes an API to read the results

both are Spring Boot apps and in production they are running inside a docker container.

I'd like to have an E2E test which:

• Runs both components
• Mocks external APIs (that engine uses)
• Queries web and check expected results
• Check logs to see if there were any errors

I know about tools like REST Assured and karate but I have two missing parts.

1. How do you normally run this services in test?
2. Is there any convenient way to analyze the logs? (i.e. check there weren't any errors)