Can the corp rez an asset on the runner's turn? If not, can they rez it on their own turn before the "when your turn begins" timing happens?

Yes and yes.

For the timing of rezing cards, please study the Timing Structure chart

For the timing of rezing cards, please study the Timing Structure chart

This. Note the symbol at the top showing when paid abilities can be performed (doesn't need to be your turn if it doesn't cost a click, which rezzing assets does not), and then look where the symbol shows up for both the corp turn and the runner's.

For the timing of rezing cards, please study the Timing Structure chart

This. Note the symbol at the top showing when paid abilities can be performed (doesn't need to be your turn if it doesn't cost a click, which rezzing assets does not), and then look where the symbol shows up for both the corp turn and the runner's.

Note that Rezzing Cards and using Paid Abilities are different things and that while most windows that allow Paid Abilities also allow Rezzing Cards, not all do.

And note that the symbol means you can rez non-ICE cards. ICE can be rezzed only when it is approached during a run (or by a card effect).