Do you have a Green Card? If you moved to the US on Sept 3rd of 2024, you would not have met the substantial presence test, therefore if you do not have a Green Card, you would be considered a non-resident alien and would need to file a 1040NR instead of a form 1040. TurboTax does not support the 1040NR, but our partner Sprintax does and can help with your non resident return.
If you are a Resident Alien, then yes, you can claim the Child Tax Credit as a Dual Status Alien, if your child meets the criteria to be for claiming the credit. Yes, you would be eligible, if you otherwise qualify, to take the Retirement Savings Credit.