So I downloaded The Cave, and started playing it. I got a couple of the trophies (Shoplifting - stealing the postcard, and Well Done - roasting the hotdog), and then real life happened, and I took a break from the game.
When I returned after some time, there was a new patch! It automatically downloaded overnight due to PS+ subscription, and I was ready to go. I continued my game, figuring out things as I went, and then returned the postcard that I carried through the whole game. No Trophy. I should also have received several others for tasks accomplished during the game - such as Royal Buffet or What You Always Wanted, but didn't get any at all.
I thought I might have done something wrong - so I started a new game, and this time looked up the trophy guide, to make sure I hadn't done anything wrong. I took the Knight, as I had done that level previously, and I didn't get the trophy, but now I knew where it was - follow the King to the Princesses bedroom at the end of the level. Well I did that, and no trophy.
So I called Playstation support. They had not had any complaints about The Cave, but they had me reset the console and do a forced filesystem cleanup, just in case. I also deleted The Cave, deleted my save file, and re-downloaded and installed it.
Next step, I started up the game, manually applied the 1.01 patch (instead of having PS+ do it for me overnight), and did exactly the same thing - started a new game, took the Knight, went to get that trophy, and still didn't get it. At this point I thought I was doing something wrong, so I looked up the Trophy Guide again, watched the video, and figured out I was doing things exactly right - I just wasn't getting the trophies.
So I uninstalled the game, wiped the save file again, turned off PS+ auto-updating, re-downloaded it a third time, and started the game up without applying the 1.01 patch. Again selecting the Knight as one of my characters, I repeated the same process as before and presto! I got the trophy! Hurray!
The next day, I continued my game, and applied the 1.01 patch accidently. While it was downloading, I figured everything would be okay, since I got a Trophy yesterday, and everything should be fine and dandy. Of course, when I got to the next section where I should get a Trophy (Fire in the Hole - 3 deaths from the miner simultaneously), suprise surprise, no Trophy. Uninstall, re-download, re-install, continue game without patching, get killed, and the Trophy is mine.
Conclusion: 1.01 does not let you get Trophies. Not sure if it's because I started under 1.0 and then patched later or if it's just the patch. Since I deleted my save game, the first scenario seems unlikely.