What is the suspended floor made of? Wood, concrete, or steel?

Wood

You need to get an electrician in to advise and quote. Not easy to retrofit unless it is a suspended ceiling.
Spacing depends on the coverage of selected down light and the ability to install in certain locations. They need a cavity to be installed into. The electrician will need to consider beams and joists that block paths for cables, if they are timber joists then they might be able to drill a hole through. There will probably be a lot of holes in ceiling that will need patching when you are through.

Thanks, Kate. I am looking to get the house painted as well. So hoping that the painter can fix the holes..

How about surface mounted lights? Will they have the same installation hassle?

Downlights need a cavity for the light and transformer to sit it, as well as your cables that connect to a wall switch. Track lighting and surfacing lights just the cabling issue. Presumably you have lights already so track lighting may be a better option.