Was the German sdkfz 250 series more or so like a recon halftrack?
If I understand your question correctly, no. It was designed to be used by the Panzergrenadier troops as a light halftrack personnel carrier (as opposed to the mid weight 251) and a specialist recon variant, the 250/9, was developed to fill a recon role.
Did they see service in Italy until the war was over?
Yep!