Points
the points depend on what you inherited from you parents(your tamagotchi) because like if you had a match maker marriage you don't know the other's skill points.
Awards
You always keep your awards because one of the main objectives for Tamagotchi Music star is to see what happens when you get the 8 awards.( I had 6/8 on first gen!

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Special Items
Special items are kept from being purchased, or gifts and what not. Its like the parent is leaving its life behind.
Food and Snacks that you bought
Its the same with the food and snacks like the special items. The kids gotta eat.
Toys
Toys are always kept. I think as a souvenir from the previous generation , or generations before. Sometimes because the toy that the previous parent had, the new baby might not like and he/she might like a toy from the previous generations.
Instruments
Its the same thing with instruments. They may not like it, and if they don't their is a selection from the previous generations.
P.S. Anytime the parents leave they take half of the money. If you are left with none, that means your tama really needed it because they didn't have much before. They will give gifts as money and food too.