I want for example to measure the diameter of a circle but I could not find how to do that.
The diameter of a circle is the same as its width (and height). So technicaly, you don't actually have to measure anything, because if you select the circle with the Select Tool S you can see its diameter displayed in the controls bar along the top.
As for snapping, you can use the snapping options with the Measure Tool. Just switch on the snapping options you need. You can snap the measure tool to the rotation centre, and the path. This will give you the radius. The diameter of a circle is twice the radius.
Another possibility is to open the XML editor Ctrl+Shift+X , and select the circle. The radius will be shown. All you need to do is double the radius to get the diameter. This is useful if you don't want to include the stroke as part of the diameter (if a stroke is applied).
Even if an object is not an SVG circle (which will not show a radius like that above), one could easily remove the stroke temporarily and check its width in the controls bar. then apply the stroke again. Or even simply subtract the stroke width from the circle width. It's just simple arithmetic