I snapped these screenshots of all the levels I can reach and pause comfortably. Each of these pics illustrates the position of all the treasures, balloons and doors. Ignore the positions of the baddies!
Round 1 - Welcome to Mappy! The first few levels are great for experimenting and maximizing your points. The cats are slow and the platforms are all uniform.
Round 2 - Again, go for points. You should be able to clear 20,000 and get your first bonus life before the end of this round.
Round 3 - The first bonus level. Although the ending of the music may stress you, it is very easy to get a perfect score, provided you make every bounce-change smoothly.
Round 4 - The first level with an attic. Be careful of venturing at that topmost level, as the long platform lets Meowkies catch up with you.
Round 5 - The Goro cat coin may start to appear on levels 4 to 6, depending on your score and time.
Round 6 - The Mappy FAQ suggests a score around 60,000 by this point. Your second extra life will come at 70,000 (if you're playing under default settings).
Round 7 - The second bonus level. Just as the first bonus round, there is plenty of time to pop all the balloons yourself. In fact, I usually finish this board with more time left than Round 3. Remember to abort a needless high-bounce by hitting a wall.
Round 8 - This level introduces the bells. Get as many cats under the bell and then let yourself bounce right into it... the bell will fall and briefly stun all the cats in its path. Each bell only works once.
Round 9 - By now, your odds of grabbing pairs of treasures for max points are growing very small.
Round 10 - Hurry warnings are coming very quickly now...
Round 11 - The third bonus level. Goro himself will pop the final balloon for you (the one in the top left corner).
Round 12 - Note the two trapdoors in the upper right and lower left. This board also introduces the dead-end platforms. You can outrun the Meowkies again on this level, until the Hurry alarm rings.
Round 13 - Watch those trapdoors! Be extra careful in grabbing the safe on the right hand side... a trapdoor is positioned in front of it on the dead end platform.
Round 14 - Now the trapdoor/dead end combo is on the left hand side.
Screenshots for the rest of the rounds will appear eventually.