[Task]: Playground frontend integration tests

### What needs to happen?

Initial URLs:
- [x] #24729
- [x] #25036
- [x] #25037
- [x] #25038
- [x] #24959
- [x] #25039

Selection:
- [x] #24730
- [x] #24779
- [x] #25051
- [x] #25052
- [x] #25053

Editing:
- [x] #24937
- [x] #24938
- [x] #24939
- [x] #24940
- [x] #24941

Running:
- [x] #24780
- [x] #24936
- [x] #24781
- [x] #24923
- [x] #24922
- [x] #24921

Code sharing:
- [ ] Sharing an unchanged precompiled example.
- [ ] Sharing changed code.
- [ ] Opening a shared code, editing it and sharing again.

Miscellaneous UI:
- [x] #24792
- [x] #24759
- [x] #24793
- [x] #24794
- [x] #24795
- [x] #24796

There are also various external links. We should investigate if it is possible to test clicks on those because the tester runs inside the page.

Automating the test cases for manual testing: https://docs.google.com/spreadsheets/d/1nKIX4_bLLRrEvWbeNLUdyT60mtd7yCMDFSYMudWyzxE
- [x] #25131
- [ ] #25132
- [ ] #25133
- [ ] #25134
- [ ] #25135
- [ ] #25136
- [ ] #25137

### Issue Priority

Priority: 3 (nice-to-have improvement)

### Issue Components

- [ ] Component: Python SDK
- [ ] Component: Java SDK
- [ ] Component: Go SDK
- [ ] Component: Typescript SDK
- [ ] Component: IO connector
- [ ] Component: Beam examples
- [X] Component: Beam playground
- [ ] Component: Beam katas
- [ ] Component: Website
- [ ] Component: Spark Runner
- [ ] Component: Flink Runner
- [ ] Component: Samza Runner
- [ ] Component: Twister2 Runner
- [ ] Component: Hazelcast Jet Runner
- [ ] Component: Google Cloud Dataflow Runner