Yes to the first question.
When you bought the house the seller had already paid the taxes for all of 2019. You reimbursed them for the time period you owned the house, so you (indirectly) paid those taxes and they are deductible for you.
Maybe to the second question.
Did you receive a 1098 statement from your lender? It should show on the statement how much property tax the lender paid on your behalf in 2019. This amount is deductible in addition to the amount you paid at settlement.
[Edited | 3/19/2020 | 12:08 pm PDT]
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