Explore how to simplify the code for managing connections to local and remote Jupyter Servers #12832
Description
### Tasks
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13610
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13578
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13577
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13631
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13632
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13633
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13488
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13345
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13112
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13015
- [ ] https://github.com/microsoft/vscode-jupyter/issues/8134
- [ ] https://github.com/microsoft/vscode-jupyter/issues/11645
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13300
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13634
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13635
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13637
- [ ] https://github.com/microsoft/vscode-jupyter/issues/13808