I also have a Lumia 920, and run both the Geocaching Live and Geocaching Plus apps. Geocaching Plus tends to be about 5-10 metres off (sorry, no clue what that is in feet...15-30, I think). I have not tried the multi caches, but I run LTE network, and with a decent data plan can likely log into the website for anything as I find information. I plan on a multi this weekend, so I will update.
As for the original question, when I have a screen open and hit the "windows" button, I am taken to the home screen. When I re-enter the app, I am at the beginning again. To get past this, I use wifi to save a cache for offline use, and then access through my offline (backpack) bookmarks. This way, it is only one extra step, rather than doing it ALL over again. I also tend to take the GPS unit with me, as I am not sure yet on the accuracy of the app's readings. One thing I REALLY dislike about the Geocaching Live app is that you cannot leave the download page while downloading a search or PQ. If you do, it is lost and will need to be re-started.
In the Geocaching Live app, you have the options of searching; nearby, by cache ID, by address, postal code or landmark, trackable code, keyword or cache name, or on a map. Downside is that any search run through the app maxes out at 30 caches/search, and also maxes out at a 35 mile radius...not so good if you are looking at going further abroad without more detailed information about a search area. However, it does also have driving directions for any cache as a search option.
Bonuses on the geocaching Plus app are: able to see caches from other sites as well; search is done vi a map, so it is more user-friendly for the technologically inept (like me) and it does not restart you at the "start screen" when you close and reopen the app. You are also able to mark a waypoint as needed for the return hike out of the middle of nowhere. (Handy if you tend to carry a cache with you, just in case you find a "perfect!" spot, or want to revisit somewhere another time) With this app, you can see cache description, any pictures, the log book, as well as details of the cache.
I think both apps are decent, although I prefer the geocaching live just for the ability to do everything on the go, as I cache...hope this helps!