Hi
It is neither necessary nor appropriate to close apps on Apple Watch in order to preserve battery life.
Apps that you have chosen to add to the Dock may update their content automatically, depending on their features. If preferred, you can remove apps from the Dock:
Switch between apps using the Dock on your Apple Watch - Apple Support
Apple's tips for maximising battery life on Apple Watch are available here:
https://www.apple.com/batteries/maximizing-performance/#watch
If an app is not working properly, it may instead help, in the first instance, to restart your watch (normally) before following any further troubleshooting steps:
Restart your Apple Watch - Apple Support
The options that you describe as being presented by your watch upon pressing the side button are correct under the latest version of watchOS - and it is not possible to change them.
Power Reserve mode can be turned on, if desired, via Control Centre. To access the setting: when viewing the time on your watch, swipe up to open Control Centre > tap on the battery percentage:
Check your battery and charge your Apple Watch - Apple Support