Edit GPX tracklogs from GPS point by point

[routeconverter]

First problem: every time I double click on a point of the tracklog that I want to move, and the little red balloon appears on the map, I need to click a third time on the baloon and drag it, for a total of 3 clicks per point.

You can reduce this to 2 clicks by clicking close (15 pixel) to the point on the map and moving it. See http://www.routeconverter.com/select-positions/en for more options

Worse, once I move a point, the map zooms all the way out and I have to zoom in again! Is there a solution for this?

Yes, there is a bug in the 2.15. Please try the prerelease, it solves the problem for me.

Second question: is it possible to enable little icons (say, little squares like on GoogleEarth), so that I can just click once and drag them?

No