I wanted to check if my android device is going into DRX mode.
[1] I was looking for an existing app that could tell, but I haven't found one..
[2] Is there a software to connect directly to the device that will measure it ?
[3] Is there a smarter way of detecting the DRX on/off it without using an app or software ?
(1) Yes, Nemo Handy for example, but see below. I would not expect to find any of these apps in the Play Store.
Regarding (2), yes you can use a software application such as Qualcomm QXDM to examine the signaling between the device and the network. There is a great deal of information with respect to DRX that can be captured with this software.
Most of this type of software is quite expensive unfortunately. I am not aware of any freeware that will capture the same information. You can find out more about QXDM here: QXDM pdf
(3) It would be much more difficult, but there may be a way to detect the state of the tranceiver programmatically.