Can you claim back property taxes that you paid this year? For example you owed for 2017 and 2016 and paid those in 2018. Can you claim that on the your 2018 taxes?

Yes, you can claim them. Whst counts is whhen it was paid.

Yes - If you paid those taxes in 2018. 

Please keep in mind that because of the new tax regulations your state and local tax deduction is limited to $10,000.  Go ahead and enter the total because it may affect your state return even if the taxes may be limited on the 1040.