Your first screenshot looks like there were two caches on top of each other. The one behind being the one that was a pending log. Please test turning filtering on and off.
This is correct functionality. The Adventure Labs API (way to get data from geocaching.com) does not provide a way to use the map bounds in a search. So when that option is turned on, no labs can be returned.
Do you have some replication steps for searching for a GC and the cache doesn't show and label shows? Do you have the option "Clears map on refresh" off?
For offline maps, do you have replication steps?
Ah, I see. Live and Offline are two separate parts of the app, so I wouldn't want them to move to match positions. Maybe there is some middle ground and I can think of a way to help your use case...
Yes, view the offline list. Then tap the ... button and choose "Add to List". Choose the caches you want then at the top switch to "Online" where you can save the caches to an already created list or create a new one.