With respect to the first part of your gripe, you need to do some prep work if you plan to use LP on the road. I got caught be an equally annoying LP issue a couple of years ago. Burned into my mind was the 'Securely access your passwords anywhere you have internet access and a web browser' boast.
So when I wanted to check my Gmail account from my nephew's house in Alaska -- far away from my home here in Michigan -- but I couldn't login because LP didn't recognize the computer I was sitting at. Of course, I knew my master password, I type it countless times every day. But LP wanted to protect me and sent an authentication code to my Gmail account -- yeah, the one I couldn't log into because I use LP to remember its password. This is equivalent to my favorite computer error message, ever: "Keyboard error. Press any key to continue." Of course, they were happy to also send the code to my cellphone, which had no service because I was out in the freakin' tundra....
I was not a happy camper...
The second part of your problem is odd. Under normal circumstances I use LP daily on up to 7 different devices, and don't have any difficulty with duplicate sites/password warnings. You're supposedly accessing the same list of sites and passwords with every device you use LP on. Something's very odd here. Of course, the way LP has degenerated of late, it's no telling what's likely to go wrong.