I paid a deposit to a contractor in Dec 2017, for work he will actually do on my rental property in 2018. Is the deposit amount claimed in 2017 or 2018?

You will claim it in the year you paid it.  So although he began work in 2018, if you paid out the money in 2017 you should include it on your 2017 tax return.

Does this also mean that if I was invoiced for work done on my rental property in Dec. 2017, but I paid the invoice in Jan.  2018, it goes against my 2018 return?

Yes, if you paid the invoice in 2018, although you received it in 2017 you should put on 2018 return, not 2017.