Base trim before carpet

Sure. Why not? If you have a brad gun with headless pins, they leave a hole that would be almost undetectable from a few feet away anyway. With a little care about where you place the pins, you might even skip the step of filling the holes, certainly so those in baseboard and possibly in ceiling trim.

For trim at eye level where it may be more critical to hide the holes though, the right color wax crayon used for filling nail heads will do wonders. You can buy them in wood colors for this purpose. Rub a little in, then clean the surface with a cloth or your finger.

IMO, a sloping piece would look awful.

Better to find something that 'explains' the transition. Normally, this wouldn't be an issue if you have a doorway or archyway between the two rooms, so that you would have an architrave that would create a vertical division between the two rooms.

If the two rooms merge, the ideal would obviously been to have discovered the difference beforehand and to cut down the height of the baseboard in the room with the higher floor to end up with a consistent baseboard height. I am assuming that you don't wish to remove any and cut it down (or build up the lower baseboard, if that's possible, for example if it is not molded then you could add additional height without removing what's there).

If it is molded, then I would cut a section of the molding and create vertical 'ends' to both the high and low sections (for example with a mitered joint, like a picture frame), making sure that the vertical pieces join neatly to both ends of the existing horizontal runs, and also to each other, thus creating a division, but one that will hopefully look sensible.