You might not be getting battle trophies because you didn't set up a designated spot to save them to at the beginning of the game.

When you're still on Hyda in the battle simulator, you will be given a chance to prepare a place to save battle trophies on your memory card. If you don't do this then you can't get any battle trophies.

Of course, this doesn't mean you have to start over. Just start a new game and get to the battle simulator part, designate a memory card to save the trophies on, and continue the game from where you left off.

Some trophies are easy to get, some are hard. All you need to do is meet the requirements and finish the battle.

As far as inventing goes, yes it sucks. The only way you can get items through IC is by either specify plan or original invention. The items that an npc inventor makes don't come with them. You can use the NPCs in your invention lines, and you'll get whatever you make using those characters, but they don't come with any items.