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Sims (even babies) get an aging glitch. Starting on the day after the baby is born, you normally see the green progress bar on the Simology tab. On the normal lifetime setting, babies age to toddlers in three days. But if that green bar isn't showing, then your baby is glitched. Since the birthday cake didn't work, you can try using testing cheats to age up the baby. With testing cheats enabled, shift-click on the baby and choose Trigger Age Transition. To turn on testing cheats, press Ctrl-Alt-C and type "testingcheatsenabled true" without the quote marks. When you're done, repeat the process and replace the word "true" with "false" to turn them back off.
As for the celebrity items, only certain things are designated in the game to be given as gifts to celebrities. The more celebrity stars your Sim has, the more expensive the items will be.