The filling can be done the day before without it having much effect on freshness. If making the filling on the day of serving, give it at least an hour to cool before adding it to the cake.

The cake will need a couple of hours to completely cool before you fill and ice it, and ideally should be made the same day you serve it.

Icing should be made then applied to the cake immediately, do that last.

So... filling the day before, then bake the cake at least 3 hours before you need to serve it. Once the cake is room temperature all the way through (check the bottom, if it's even slightly warm it needs more time) you can add the cold or room temperature filling. Then ice the cake.

A ring of icing around the edge of the cake layers as shown in this video will keep the filling from oozing out and wrecking your icing.