Yes, Lovefilm supplies DVDs that are Region 2 coded. As to whether it's the disks or the player, its sounds to me like the player may be somewhat "fussy" about reading certain disks. Maybe using a cleaning disk might help. I have found that rented or public-library DVDs can be dirty and/or scratched, so maybe look at the disks for smears, marks or scratches? I prefer a separate DVD player to one built in to the TV personally because of issues like this. A cheap separate DVD player costs around £20 now.
The number of times you can change the region setting on a computer's DVD drive is stored in the drive itself, in what is called the "firmware". You usually get 5 changes after which it is locked. You can often "flash" (update) the firmware to either reset the counter back to zero or alternatively set the region code to RPC1 (region free). If you know the make and model of your laptop drive you can Google for a firmware download (and the updater program). Or search/browse at rpc1.org. Be warned that although this is not hard for a tech-savvy person, not doing it right can permanently disable the DVD drive and make it necessary to buy a new one!
Before doing anything like that, you may be able to use a free of charge media player app called VideoLAN, (or VLC Media Player) which is inherently region-free, and is said to play any DVD. Available for Windows, Linux, Mac.