Climb Pro on the go without course?

No.

Not that I'm aware of. Not even sure if it's technically possible with current Connect IQ.

The original ClimbPro used the elevation in the course file.

Without a course, it has to kind of guess where you are going on the map. It basically creates a short look-ahead course and use that to get the elevation from the elevation (DEM) map. 

That can’t be done through IQ.

Climb pro does work with navigation though - ie if you’re out, search for a shop, it plots a route. If that route contains a gradient that constitutes a climb, it will display on ClimbPro. I was quite pleasantly surprised by this.

Yes, the device can use the local elevation (DEM) map to look up the elevation for the points along the calculated route. This is fairly easy to do.

I think that it's a shame that it doesn't have this (I think the 1040 does). Arguably it's quite a handy feature for a unit used by folks that go out and explore (potentially more so than some of the power meter and V02 stuff the Explore 2 does have) but I guess that it's down to Garmin to decide what features to cut from what I suppose it a fairly budget unit.