Your inventory is where you'll find equipment and rewards from missions across all game modes.
Star fragments are pieces that can be used to redeem either a 2-star or 3-star character respectively. You can receive these as rewards for completing missions in various game modes.
Allowances are bundles of coins that correlate to different amounts generated by Will in the Rewards section. For example, a "Small Allowance" is equal to two hours worth of coins. Similarly, Test Tubes work the same way, only with Goo as the reward.