Shift lever and rear derailleur compatibility

For readers, the ST-EF505-R is a 9 speed shifter with hydraulic brake lever combined. The cable pull on 8 and 9 speed is the same, so you will be able to use the ST-EF505-R on your bike.

While there is nothing inherently wrong with combined units, personally I prefer a separate brake and shifter. If you are buying disc brakes it may work out cheaper to by a complete brake set and then a suitable shifter, so its an option certainly worth considering.

I suggest before proceeding familiarize yourself with the many discussions here and elsewhere about disc brake upgrades, in terms of both difficulties and cost. Properly setup quality V brakes with good pads can be very effective.

Yes, you can use any Shimano 8 speed shifter with a Shimano 8 speed derailleur and cassette.

Generally Shimano (and SRAM) used the same sprocket spacing for all cassettes with the same number of sprockets (more sprockets means narrower spacing though). All MTB shifters up to 9 speed use the same cable actuation ratio (ratio of cable pulled to cage movement), and all shifters pull the right amount of cable per shift for the number of speeds they are for.