If you want your cat to stay with you longer you have to raise at least one of your stats to level four(that will put you as an intermediate owner) the girl with blue hair will let the next cat you own stay with you for 21 days. Im not sure what level you have to be an expert, but you have to get at least one of your stats to a level larger than 5.
If you want to get your punch level higher then buy the punch bag at the shop listed under toys. use it at least 3 to 6 times a day(this will get the stat higher faster) if you have 14 days with your cat. But of course theres the looks stat to worry about when you use it. If you dont want to use that then use the bouncy ball it raise your punch friend and mood stats and wont damege your looks stat.
If you want to get your looks stat higher before the deadline use the brush at least 3 times a day or give it a lot of water. Of course you want to raise your other stats but you dont want to lower your looks stat use the bouncy ball. Another way is to keep using 'pat head' in "bond".
Feed your cat seafood jerkey under "snacks" before you feed it food when you have one bar left in "status" for hunger. PLay with bouncy ball. Use 'pat head' in 'bond'.
Keep using pat head and you will get a new gesture and keep using those gestures to earn another one.
You can go to the store at any time. When you are there time stands still.
If you get your Catz punch level very high, when the girl comes over on the last day she will say, "Oh my goodness. Your Catz is super strong. You are now at the Amateur level". Every time your punch gets very high, you will be advanced to the next level on the last day. The levels are: Amateur, Advance, Expert, Master, and Supreme Cat Owner.
If you want to get higher level you take a nap every sunday. but before you do that you have to make sure your at least a "level 2" friend with your cat. also give it lots of snacks before you take a nap. lso shut your door on the first 2 days.