I had a play with a damage calculator I found online and my conclusion is: there are way too many variables to produce a definitive answer: base stats, IVs and Nature all play a part, even at level 5.
If your Pokemon has perfect IVs and beneficial Nature and your opponent has 0 IVs and negative Nature, it looks like you could deal at least an extra 20% damage than if those stats were reversed. I tried a good Snivy against a bad Tepig and got 42% damage from using Tackle, whereas a bad Oshawott against a good Snivy only did 14% damage with Tackle.
In my experience, using any status move is not worthwhile. It always takes the same number of moves before your opponent is KO'd. Example: assume an average damage of 20%, that would normally take 5 Tackles to KO your opponent. If you used Leer first, Tackle would deal ~30% damage (20*1.5). Now it takes Leer + 4 Tackles, so you don't gain anything.
Also note that your opponent uses moves randomly, which you can benefit from: Tackle>Leer>Tackle always does less total damage than Leer>Tackle>Tackle.
My advice: never use more than one status move, and if you do use one always use it as the first move. Then just spam Tackle.
Your research is spot on. Any NTR code DS game will work on any 3DS. So, why doesn't it? The most likely reason is that it's a fake/copy/bootleg. These are quite common, especially for popular brands like Pokémon. If you post pictures of the cart (both sides), box front and back and manual, I might be able to tell you if it is indeed one.
Looking at your specific cart: that looks legit indeed. It's a European copy though, so your 3DS actually has the correct region code anyways.
It might actually be damaged then, even if it works on the DS. You could try cleaning the contacts with a Q-tip and alcohol.
Best Answer
They're nothing like the main Pokemon games. In the Mystery Dungeon games, you are exploring dungeons and levelling pokemon and going on quests. Pokemon Ranger is more about collecting Pokemon (but not using them in teams or for fighting or anything) and solving puzzles and completing tasks.
Neither of those series really follow each other in any way - every Mystery Dungeon or Pokemon Ranger game can be played as a standalone title, and you won't be missing out on any story or anything. Each game will explain any controls/story/etc elements that you might need to know.